Revisiting the works of Mihail Benado

Sergiu Rudeanu, Dragos Vaida

University of BucharestFaculty of Mathematics and Informatics

Str. Academiei No. 14, 010014 Bucharest, Romaniasrudeanu@yahoo.com,corresponding author; dvaida@rdslink.ro

May 20, 2010

Abstract

Mihail Benado was a Romanian algebraist who worked in lattice the-ory, group theory and interconnections. He is the creator of the theoryof multilattices, which nowadays has important applications in computerscience. The present paper tries to sketch an overview of Benado’s works,which comprise not only multilattices, but several other important lines ofresearch, which deserve to be recalled to the contemporary mathematicalcommunity.Key words: multilattice, regular product, Schreier refinement theorem,Dedekind quadrilateral, division semigroup, partially additive semantics,monotone connection, semidistance, diametric spaceMSC 06-02, 06A06, 06B99

Mihail Benado was a Romanian algebraist who worked in lattice theory,group theory and interconnections. He is the creator of the theory of multi-lattices, which nowadays has important applications in computer science. Thepresent paper tries to sketch an overview of Benado’s works, which comprise notonly multilattices, but several other important lines of research, which deserveto be recalled to the contemporary mathematical community.

Benado1 was born on July 5, 1920 in Bucharest. He was a scholar at the ele-mentary school of the Jewish Community, then at high school “Matei Basarab”,from which he graduated in 1940. Because of the anti-Semitic law “NumerusClausus” in vigour in Romania of those days, Benado could not become a stu-dent of a State University, but he attended the courses of a private university.In 1944 he became a student of the Faculty of Sciences (including divisions ofmathematics, physics and chemistry) of the University of Bucharest from whichhe graduated in 1948, the very year in which the Faculty of Mathematics and

1The biographical data of Benado are taken from the monograph by G.St. Andonie [GSA],vol.III, §36.

1

Physics was created as a separate faculty. From 1948 to 1962 Benado held teach-ing positions at this faculty, and from 1950 to 1962 he was also a researcher atthe Institute of Mathematics of the Romanian Academy. In January 1962 heresigned both positions.

Unfortunately, in the sixties Benado gradually isolated himself from themathematical community of Romania and, except a rather short period in whichhe was with the 3-year Pedagogical Institute of Bucharest, there are no infor-mations about Benado’s life after 1964, except that he emigrated to Israel.2

Mihail Benado was a pupil of Dan Barbilian, who directed Benado’s interesttowards group theory and lattice theory, the latter being at that time a youngbranch of mathematics. The fact that Barbilian was both an algebraist and ageometer had a strong influence in the scientific career of Mihail Benado.

Two major research themes of Benado were the theory of regular productsof operator groups initiated by Oleg Golovin and the Schreier-Zassenhaus andJordan-Holder refinement theorems in group theory and their generalizations tomodular lattices. Benado introduced new concepts related to regular products,generalizing several results of Golovin and generalized this theory to the lattice-theoretical level. The main concern (which can be detected even in his firstpaper [1] dealing with Galois theory) of Benado’s ample contribution to thefield of refinement theorems was the passage from modular lattices to arbitrarylattices and even to partially ordered sets and to his multilattices.

The theory of multilattices is in fact the most important creation of MihailBenado. He has relaxed the concept of a lattice by requiring only the existenceof certain minimal upper bounds and of certain maximal lower bounds insteadof the least upper bound and the greatest lower bound.

The spirit of multilattices is also present in other papers of Benado. Thus, heintroduced the monotone connections of types I, II and III, which are similar tobut different from the conventional Galois connections, and characterized semi-lattices and semimultilattices in terms of these monotone connections. Benadoalso generalized several results of Glivenko and Barbilian on metric lattices. Theworks of Mihail Benado comprise two other big constructions. One of them isthe theory of diametric spaces, an attempt to develop an axiomatic theory of thedistance between two bodies (Korper). The other construction is a general the-ory of partially ordered sets, even more general than the theory of multilattices.Unfortunately, these two projects have remained unfinished.

Mihail Benado, like his master Dan Barbilian, had a complex and rathercontradictory human personality. To a certain extent, this had consequenceson his work. Thus e.g. he has no written course, although he was an excel-lent teacher. In particular the mathematical orientation of the present authorshas been strongly influenced by his teaching and research3 Also, we have noinformation about a possible PhD obtained by Benado, although – as we hopeto prove in this paper and a next one – he was a highly inventive author ofmathematical papers. Benado wrote no book, yet he is the author of several

2Yet the paper [58] still mentions Benado’s address in Bucharest.3In particular Benado used his paper [16] as the support of a one-semester free course

taught to the second author.

2

comprehensive theories which deserve to be better known by the mathematicalcommunity.

The papers of Mihail Benado are written in French (31), Romanian (17),German (8) and Russian (3). He has no paper in collaboration, but he corre-sponded with several mathematicians. Among them the most active were JanJakubık and Milan Kolibiar, who proved the existence or non-existence of math-ematical objects having certain properties that interested Benado and pointedout a few minor gaps subsequently corrected by Benado; cf. footnotes in papers[24], [32], [38], [51], [53], [54]. On the other hand, the first works on multilatticesof Jakubık and Kolibiar appeared starting with the years 1956-1958, as shown inthis paper. A full report on the subsequent development of multilattice theoryand its applications is planned for a future paper of ours.

The present paper is structured as follows. Part A: theory of regular productsof groups. Part B: theory of Schreier-Zassenhaus and Jordan-Holder theorems.Part C: multilattices and applications, with a link to contemporary applicationsin computer science. Part D: the other papers of Benado. Part E containstwo bibliographies: the papers of Mihail Benado4and a selective bibliographyof the sources of Benado. Part E is followed by paragraphs Further research,Acknowledgements and the references of the present paper.

Parts A, B and C can be read independently of each other. Part D relies onPart C.

We refer to the papers of Benado by simple numerical codes [1], . . . ,[59].The sources of Benado are cited in the form Author [year], e.g. O.N. Golovin[1950], [1951a,b]. The other citations are listed as “References” under suggestiveacronyms; for instance, [Bir] and [Bir67] indicate a paper and a book by G.Birkhoff.

PART A

The theory of regular products of groups

In this Section we survey the papers [21], [30], [31], [32], whose main resultswere announced in [25], [27], [28], [29], and the papers [37], [41]. In this seriesof articles M. Benado generalizes the theory of regular products of groups, in-troduced by O.N. Golovin [1950], [1951a,b]; cf. O.N. Golovin and N.P.Goldina[1951].

The works of Golovin and Benado refer to operator groups and permittedsubgroups; yet for the sake of convenience in the sequel we refer simply to groupsand subgroups.

We begin with some notation and other prerequisites.4To the best of our knowledge, the list of Benado’s papers is complete and he has written

no book.

3

Let G be a group. The subgroup generated by a family Gii∈I of sub-groups is denoted by

∨i∈I Gi. The singleton subgroup is denoted by E. If H is

a subgroup, the notation H means the normal subgroup generated by H. Thecommutator of two elements x, y is the element xy = x−1y−1xy. The commu-tator of two subgroups K,H of G is the subgroup K H generated by all thecommutators k h with k ∈ K and h ∈ H. The commutator K H is a normalsubgroup of K∨H; in particular GH is a normal subgroup of G. For any sub-groupH of G, by setting G1H = GH , Gn+1H = G(GnH) (n = 1, 2, . . . ),one obtains a decreasing sequence G 1 H ⊇ · · · ⊇ G n H ⊇ . . . of normal sub-groups of G.

Now let G = G1 × · · · × Gn be a finite direct product of groups. For eachi = 1, . . . , n, let Hi = (g1, . . . , gn) ∈ G | gj = e for j 6= i. It is easy to see thatthe His are normal subgroups of G which generate G and for each i = 1, . . . , n,the intersection between Hi and the subgroup of G generated by the other Hjsreduces to E. Therefore, following A.G. Kurosh [1953], a group G is a directproduct of a family Gii∈I of subgroups, written

G =∏i∈I

Gi ,

provided the following conditions are satisfied:each Gi is a normal subgroup of G ;G =

∨i∈I Gi = ;

(∀i ∈ I) Gi ∩∨

j∈I−iGj = E .O.N. Golovin has introduced the following more general concept. A group

G is a regular product of a family Gii∈I of subgroups, written

G =⊗i∈I

Gi

provided the following conditions are satisfied;G =

∨i∈I Gi ,

(∀i ∈ I) Gi ∩∨

J∈I−iGj = E.Golovin has also defined the first normalized commutator of the subgroups

Gi asK1 =

∨i,j∈I;i 6=j

(Gi Gj) ,

and the n-th step normalized commutators are defined inductively:

Kn+1 = G Kn (n = 1, 2, . . . ) ;

they form a decreasing sequence of normal divisors of G.Following Benado, a subgroup H of a group G is called an n-step normal

subgroup if G n H ⊆ H. The conventional normal subgroups coincide with the1-step normal subgroups, but for n > 1 there exist n-step normal subgroups

4

that are not normal subgroups. Clearly each n-step normal subgroup is also an(n + 1)-step normal subgroup. For each n, the n-step normal subgrups form acomplete sublattice of the lattice of all subgroups of G.

Also, a regular product⊗

i∈I Gi is called by Benado an n-step direct product,written

(n)⊗i∈I

Gi ,

provided its factors are n-step normal subgroups: G n Gi ⊆ Gi (i ∈ I).Here are a few sample results linking these concepts [31].A regular product is an n-step direct product if and only if Kn = E. The

n-step direct product is associative, to the effect that if G =⊗(n)

i∈I Gi and eachGi =

⊗(n)j∈Ji

Gij (i ∈ I), then Gi =⊗(n)

i∈I,j∈JiGij . The subgroup property says

that if G =⊗(n)

i∈I Gi and Hi are subgroups of G such that Hi ⊆ Gi (i ∈ I), then∨i∈I Hi =

⊗(n)i∈I Hi. A non-trivial free group cannot be expressed as an n-step

direct product. If G = G1 ∨G2 and G1 (or G2) is an n-step normal subgroup,then Kn ⊆ G1 (or Kn ⊆ G2).

The above theorems (previously known for n = 1) and other similar resultsgeneralize theorems due to Baer, Levi and Golovin, and solve a problem raisedby Mal’cev and Golovin.

Benado also found [21] a condition equivalent to the complete associativityof a regular product, meaning that if G =

⊗i∈I Gi then for any partition

I = I ′ ∪ I ′′, I ′ ∩ I ′′ = ∅ we have G = (⊗

i∈I′ Gi)⊗ (⊗

i∈I′′ Gi).On the other hand, the concept of a direct product of subgroups can be lifted

to a lattice-theoretical level, as Kurosh first did. He introduced the concept of acompletely modular latice, meaning a complete lattice L which satisfies the fol-lowing strengthening of the modular law: for every two families xii∈I , yii∈I

of elements of L, if xi ≤ yj for all i, j ∈ I with i 6= j, then

(∨i∈I

xi) ∧ (∧i∈I

yi) =∨i∈I

(xi ∧ yi) .

Taking a two-element set I and y1 = x1 ∨ x2, one sees that every completelymodular lattice is indeed modular, but there exist modular complete latticesthat are not completely modular. Kurosh defined direct decompositions5

a =.∨

i∈I ai

in a completely modular lattice, meaning that the following conditions are ful-filled:

a =∨

i∈I ai ,(∀i ∈ I) ai ∧ a∗i = 0, where a∗i =

∨j∈I−i aj .

So a direct product of groups∏

i∈I Gi is a direct decomposition of the greatestelement G of the completely modular lattice of normal divisors of G.

5which he rather improperly called direct sums

5

The endomorphisms of a direct decomposition 1 =.∨

i∈I ai of the unit 1 of Lare the mappings ϕi : L −→ L defined by

ϕi(x) = ai ∧ (x ∨ a∗i ) (i ∈ I) .

These functions are in fact just order-preserving maps, but when applied to thelattice of normal subgroups they yield a system of group endomorphisms.

In order to study regular products of groups, Benado uses a similar tech-nique, but unlike Kurosh, his axiomatics has in view the lattice of all subgroupsof a group, which is not modular. So, in his lattice-theoretical approach, Benadoworks with a complete lattice in which he needs a closure operator as a modelfor the passage from a subgroup to the normal subgroup generated by it. Basedon works by O. Ore, V. Korınek and D. Barbilian, Benado [30] endows L with abinary relation N subject to the following axioms. Whenever aNb one requiresthat a ≥ b and

a ≥ x ≥ y imply x ∧ (b ∨ y) = (x ∧ b) ∨ y ,a ≥ y ≥ b and a ≥ x imply y ∧ (x ∨ b) = (y ∧ x) ∨ b ,a ≥ x implies xN(x ∧ b) ,a ≥ x and xNy imply (b ∨ x)N(b ∨ y) :

further, 1N1, 1N0 and the set LN = a ∈ L | 1Na is a complete sublatticeof L. The lattice L is then called a Lie lattice. The idea of this axiomatizationis that L should represent the lattice of subgroups of G, the relation ANBmeaning that B is a normal subgroup of the subgroup A, the element 1 ∈ Lrepresents the group G, so that LN is an abstract version of the sublattice ofnormal subgroups of G. It should be noted that LN is a modular lattice bythe first axiom above, but need not be a completely modular lattice. Otherinstances of Lie lattices are the lattice of subalgebras of a Lie algebra and thelattice of subrings of a ring [27]; in the latter case ANB means that B is atwo-sided ideal of the subring A.

Benado works with the closure operator associated with LN . So he defines[30] a regular product

a =⊗

i∈I ai

by the following conditions:a =

∨i∈I ai ,

(∀i ∈ I) ai ∧ (a ∧ a∗i ) = 0 .The Fitting endomorphisms associated with a regular product are the orderendomorphisms αi : L −→ L (i ∈ I) defined by

αi(x) = ai ∧ (x ∨ (a ∧ a∗i )) (i ∈ I) .

The Fitting endomorphisms characterize the regular product [30] to the effectthat they satisfy six conditions with the property that conversely, any systemαii∈I of order endomorphisms satisfying these conditions is the system of Fit-ting endomorphisms of the regular product a =

⊗i∈I ai with ai = αi(1) (∀ i ∈

I).

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The results concerning regular products of groups are generalized to thelattice-theoretical level. Thus e.g. if 1 =

⊗i∈I ai and ai =

⊗j∈Ji

aij (i ∈ I)then 1 =

⊗i∈I,j∈Ji

aij ; if 1 =⊗

i∈I ai and bi ≤ ai (i ∈ I) then settingb =

∨i∈I bi one obtains b =

⊗i∈I bi; etc. Benado concentrates on the so-

called refinements of two regular decompositions. Here is a sample result [30].Consider two regular products 1 =

⊗i∈I ai and 1 =

⊗j∈J bj with associated

Fitting endomorphisms αii∈I and βjj∈J , respectively. Then the followingconditions are equivalent for two refinements ai =

⊗j∈J aij (i ∈ I) and bj =⊗

i∈I bji (j ∈ J) : for all i ∈ I and j ∈ J ,

(I) αi(bji) = aij & βj(aij) = bji & aij = bji ;

(II) αiβj(1) = βjαi(1) .

Moreover, if these conditions hold, then for every i ∈ I and j ∈ J ,

(III) αiβj(1) = aij & βjαi(1) = bji .

A variant of this result [32] says that relations (III) hold if and only if

(IV)∨

i∈I,j∈J(αiβj(a∗i ) ∨ βjαi(b∗j )) = 0 ,

where a∗i =∨

k∈I−i ak and b∗j =∨

k∈J−j bk. Besides, (IV) implies αi(bji) =aij and βj(aij) = bji for all i ∈ I and j ∈ J .

Another tool used by Benado in the abstract theory of regular productswas the concept of a relative closure operator (operateur de fermeture relatif),meaning a map which associates with every pair x, y satisfying x ≥ y, an elementy−x subject to the following conditions:

(FR1) x−x = x ,

(FR2) x ≥ y =⇒ (y−x)−x ≤ y−x ,

(FR3) x ≥ y ≥ z =⇒ y−x ≥ z−x ≥ z−y .

It is easily seen that these properties imply

x ≥ y =⇒ x ≥ y−x ≥ y ,

x ≥ y =⇒ (y−x)−x = y−x .

So, for a fixed x the map −x is a closure operator on the set (x] of all lowerbounds of x.

For instance, if − is a Kuratowski closure on a distributive lattice, theny−x = x∧y is a relative closure operator. Note that the elements a∧a∗i , occurringin the definitions of a regular product and of the Fitting endomorphisms andlargely used by Benado in the theory of regular products, are in fact the relativeclosures (a∗i )

−a. Another relative closure operator which occurs here is H−G =(G n H) ∨H.

7

Benado has also developed [27], [28], [29], [32], [37], [41] a more comprehen-sive theory of Lie lattices endowed with a commutating function (“commuta-trice”), that is, a function q(x, y) obeying a list of 12 axioms which abstract theproperties of the n-step commutator H n K of subgroups. As a matter of fact,the ultimate goal of Benado in this series of papers refers to groups. So he paysattention to lattices of subgroups equipped with a function q(H,K) satisfyingthe axioms of a commutatrice. A subgroup H of a group G is called q-normalprovided q(G,H) ⊆ H, while a regular product

⊗i∈I Gi is said to be q-regular

provided all the factors Gi are q-normal. We conclude this Section with thefollowing sample theorem [37], which generalizes several previous theorems. IfG =

⊗i∈I Gi is a regular product for which there exists i0 ∈ I such that all

the factors Gi with i 6= i0 are q-normal, then Gi0 is q-normal too, so that theproduct is q-regular.

Part B

Schreier-Zassenhaus and Jordan-Holder theorems inarbitrary lattices

There is a huge literature around the Schreier-Zassenhaus and Jordan-Holdertheorems in group theory and their generalizations to modular lattices and be-yond. One of the major research lines of Benado refers to this field, which maybe characterized as dealing with refinements of chains. The main concern ofBenado’s work in this area was the passage from modular lattices to arbitrarylattices [5], [10], [13], [14], [20] and even to posets [13] and multilattices [17],[23]. The results were announced in [2], [3], [6], [7], [9], 11] and, with emphasison groups, in [18], [19], [22]. We can only sketch his ample contributions and todo this we begin with a short presentation of the modular lattice background.

The two theorems in question are related to another well-known result, whichstates that in a modular lattice, every two intervals of the form H = [a, a ∨ b]and K = [a ∧ b, b] are isomorphic. This is proved by using the isotone maps+ : H −→ K defined by x+ = b ∧ x and − : K −→ H defined by y− = a ∨ y,which we call the canonical maps associated with the conjugate or transposedintervals H and K. Modularity implies that for every x ∈ H and every y ∈ K,

x+− = a ∨ (b ∧ x) = (a ∨ b) ∧ x = x ,

y−+ = b ∧ (a ∨ y) = (b ∧ a) ∨ y = y .

These identities show that the canonical maps establish an order isomorphismbetween H and K, hence a lattice isomorphism, too.

Benado proves the following converse of the above theorem: if in a lattice Lthe canonical maps satify x+− = x (∀a, b ∈ L) (∀x ∈ H) or y−+ = y (∀a, b ∈L) (∀x ∈ H), then the lattice is modular.

8

Indeed, note first that the canonical maps can be defined and are isotone inevery lattice. Now suppse e.g. the identity y−+ = y holds. Take a, b, c ∈ Lwith c ≤ b. Then

b ≥ b ∧ (a ∨ c) ≥ (b ∧ a) ∨ c ≥ a ∧ b ,

hence (b ∧ a) ∨ c ∈ K, therefore

(b ∧ a) ∨ c = ((b ∧ a) ∨ c)−+ = (a ∨ c)+ = b ∧ (a ∨ c) ≥ (b ∧ a) ∨ c ,

so that (b ∧ a) ∨ c = b ∧ (a ∨ c), proving that L is modular.So every lattice in which the canonical maps asociated with a, b ∈ L es-

tablish an isomorphism is modular. Note that this result cannot be provedby observing that in the pentagon non-modular lattice o, a, b, c, u we have[a, a∨ b] = a, c, u and [a∧ b, b] = o, b, because the intervals H and K of theentire lattice may be different from the above.

Now let us come to the problem we have announced, i.e., the refinement ofchains. We deal with finite chains x0 ≤ x1 ≤ · · · ≤ xn of a modular lattice L.The intervals [xi−1, xi] (i = 1, . . . , n − 1) are called the factors of the chain (aterm reminiscent of the quotient or factor groups Gi/Gi−1 of a normal chainG0 ⊆ G1 ⊆ · · · ⊆ Gn dealt with in group theory). A factor [xi−1, xi] is calledproper if xı−1 6= xi. The length of a chain is the number of its proper factors.

A refinement of a chain a0 ≤ a1 ≤ · · · ≤ ar is a chain c0 ≤ c1 ≤ · · · ≤ ctsuch that a0 = c0, ar = ct and a0, a1, . . . , ar ⊆ c0, c1, . . . , ct. If the latterinclusion is strict, the refinement is called proper. The Zassenhaus refinementsof two finite chains having common extremities, that is,

(1.1) a0 ≤ a1 ≤ · · · ≤ ar ,

(1.2) b0 ≤ b1 ≤ · · · ≤ bs ,

(1.3) a0 = c0, ar = bs ,

are the chains aij and bji defined by

(2) aij = ai ∧ (ai−1 ∨ bj) (i = 1, . . . , r; j = 0, 1, . . . , s) ,

(3) bji = bj ∧ (bj−1 ∨ ai) (i = 0, 1, . . . , r; j = 1, . . . , s) .

This makes sense because, as can be easily checked,

(4) a0 = a10 ≤ · · · ≤ ai−1 = ai0 ≤ ai1 ≤ · · · ≤ ais = ai ≤ · · · ≤ ars = as ,

(5) b0 = b10 ≤ · · · ≤ bj−1 = bj0 ≤ bj1 ≤ · · · ≤ bjr = bj ≤ · · · ≤ bjs = bs .

Another property is that for every i ∈ 1, . . . , r and every j ∈ 1, . . . , s,there are two isotone maps

ϕ : [ai, j−1, aij ] −→ [bj, i−1, bji] ,

9

ψ : [bj, i−1, bji] −→ [ai, j−1, aij ] ,

defined byϕ(x) = bj ∧ (ai−1 ∨ bj−1 ∨ x) ,

ψ(y) = ai ∧ (ai−1 ∨ bj−1 ∨ y) ,

respectively. Indeed, set ai−1 = m ≤ ai = M and bj−1 = n ≤ bj = N . Then

bj, i−1 = N ∧ (n ∨m) ≤ N ∧ (m ∨ n ∨ x) = ϕ(x) .

But x ≤ aij = M ∧ (m ∨N), therefore

ϕ(x) ≤ N ∧ (m ∨ n ∨ (M ∧ (n ∨N))) ≤ N ∧ (M ∨ n) = bji

and similarly ψ(y) ∈ [ai, j−1, aij ].As a matter of fact, the Zassenhaus refinements and the existence of the

isotone maps ϕ and ψ are valid in arbitrary lattices. If the lattice is modular,there exist rs isomorphisms

(6) [ai, j−1, aij ] ≈ [bj, i−1, bji] (i = 1, . . . , r; j = 1, . . . , s) .

To prove this we fix i and j. Using the above notation, we observe that

M ∧ (m ∨N) ≤ x ≤M ∧ (m ∨N) ,

from which we infer

ψϕ(x) = M ∧ (m ∨ n ∨ (N ∧ (m ∨ n ∨ x))) = M ∧ (m ∨ n ∨N) ∧ (m ∨ n ∨ x)

= M ∧ (m ∨N ∨ x) ∧ (m ∨ n ∨ x) = M ∧ (m ∨ n ∨ x) = (M ∧ (m ∨ n)) ∨ x = x

and similarly ϕψ(y) = y.At this point the following definition is necessary. Two chains C and C ′

are said to be isomorphic if they have the same length and there is a bijectionbetween the proper factors of C and the proper factors of C ′ such that theproper factors which correspond to each other are isomorphic.

Now the above results yield the following corollary, known as the Schreier-Zassenhaus theorem for modular lattices: every two finite chains with commonextremities have isomorhic refinements.

This results from the following two remarks. The factors of the chains (4)and (5) are the intervals ocurring in (6) plus the improper factors [ais, ai+1, 0] =[ai, ai] and [bjr, bj+1, 0] = [bj , bj ]. Each of the rs pairs of factors occurring in(6) consists either of two improper factors or of two isomorphic proper factors.So the number of proper factors is the same for the chains (4) and (5), and therequired bijection is the restriction of (6) to the proper factors.

The next theorem requires the following definition. A strict chain a0 < a1 <· · · < ar is called maximal if it has no proper refinements. It is easy to see thatthis happens if and only if for each i = 1, . . . r, the element ai−1 is covered byai, that is, [ai−1, ai] = ai−1, ai.

10

It should be mentioned that a modular lattice may have no maximal chains.However, if maximal chains do exist then the following properties hold.

If C0 : a0 < a1 < · · · < ar is a maximal chain and C ′0 is a Zassenhaus

refinement of C0, then C ′0 is not a proper refinement, hence it must be of the

form

. . . (ai−1) = ai0 = ai1 = · · · = ai, k(i) < ai, k(i)+1 = · · · = ais(= ai) . . .

for i = 1, . . . , r. It follows that C ′0 is a maximal chain isomorphic to C0.

We are now in a position to prove the Jordan-Holder theorem for modularlattices: if maximal chains do exist, then every strict chain can be refined to amaximal chain and all maximal chains have the same length.6

Indeed, let C be a strict chain and C ′, C ′0 the Zassenhaus refinements of C

and the maximal chain C0, respectively. Then the proper factors of the maximalchain C ′

0 are two-element chains, and since C ′ is isomorphic to C ′0, the proper

factors of C ′ are also two-element chains, therefore by deleting from C ′ theredundant terms we obtain a strict chain C ′′ which is still a refinement of Cand whose prime factors are two-element chains, therefore C′′ is maximal.

For the second claim suppose that C is also a maximal chain. Then C andC0 are isomorphic to their Zassenhaus refinements C ′ and C ′

0, and the latter areisomorphic. But the isomorphism of chains is clearly an equivalence relation,therefore the strict chains C : a0 < a1 < · · · < ar and C0 : c0 < c1 < · · · < csare isomorphic, which implies r = s.

Benado proves the following Schreier-Zassenhaus theorem which holds inarbitrary lattices. Suppose the chains (1) satisfy the isomorphisms

(7.1) [ai, ai ∨ bj ] ≈ [ai ∧ bj , bj ] ,

(7.2) [bj , ai ∨ bj ] ≈ [ai ∧ bj , ai] ,

for i = 1, . . . , r−1 and j = 1, . . . , s−1. Then their Zassenhaus refinements (2),(3) have the same length and the isomorphisms (6) hold. Besides,

(8.1) [ai, j−1, ai, j−1 ∨ bj, i−1] ≈ [ai, j−1 ∧ bj, i−1, bj, i−1] ,

(8.2) [bj, i−1, ai, j−1 ∨ bj, i−1] ≈ [ai, j−1 ∧ bj, i−1, ai, j−1] ,

for i = 1, . . . , r − 1 and j = 1, . . . , s− 1.Also, Benado obtains the following Jordan-Holder theorem for arbitrry lat-

tices. Suppose the chains (1) are strict, satisfy (7) and for every i = 1, . . . , r−1and every j = 1, . . . , s− 1 there are no elements a ∈ [ai−1, ai] and b ∈ [bj−1, bj ]satisfying simultaneously [a, a ∨ b] ≈ [a ∧ b, b] and [b, a ∨ b] ≈ [a ∧ b, a]. Thenr = s and the intervals of the two chains are isomrphic (in a certain order).

Another major concern of Benado was the theory of normality in arbitrarylattices, initiated by O. Ore. An element b is called α-normal if

x ≤ z =⇒ x ∨ (b ∧ z) = (x ∨ b) ∧ z ;6Unlike what happens for groups, in a lattice the isomorphism of two maximal chains

having the same length is trivial, because the proper factors are two-element chains.

11

an element a is said to be β-normal if

a ≥ z =⇒ a ∨ (y ∧ z) = (a ∨ y) ∧ z ,

and an element which is both α-normal and β-normal is called seminormal.Benado studied in great detail these concepts, as well as several new related

concepts of normality introduced by himself, for instance a concept called binor-mality such that seminormal =⇒ binormal =⇒ β-normal. With each of theseconcepts a corresponding concept of chain is associated, called α-normal chain,β-normal chain, seminormal chain, etc. Benado’s endeavour was to establishSchreier-Zassenhaus and Jordan-Holder theorems for the various types of chains.

For example, an interval [x, y] is called α-normal if x is an α-normal elementof the sublattice (y] = z ∈ L | z ≤ y. A chain (1.1) or (1.2) is said to beα-normal if all of its proper intervals are α-normal. Then the correspondingSchreier-Zassenhaus theorem states the following properties of two α-normalchains with common extremities (1): the Zassenhaus refinements have the samelength, the isomorphisms (6) hold and each aij is α-normal in [ai, j−1, ai] andeach bji is α-normal in [bj, i−1, bj ]. Further, an interval [x, y], where x < y,is called α-normal-prime if among the elements α-normal in [x, y] there is noelement z such that x < z < y. The corresponding Jordan-Holder theoremstates that if all the intervals of two α-normal chains with common extremities(1) are α-normal prime, then r = s and the intervals of the two chains areisomorphic (in a certain order).

The paper [13] proposes an axiomatic approach which passes from the var-ious normality relations such as α-normality, β-normality, etc., to an abstractnormality N , the problem under investigation being: what conditions can be im-posed upon N such as to force the validity of the Schreire-Zassenhaus theorem?Also, Benado obtains generalizations to posets [14] by creating a technique basedon Dedekind quadrilaterals (α, a, b, ω), which mean α ≤ a ≤ ω and α ≤ b ≤ ω,thus generalizing (a ∧ b, a, b, a ∨ b). For the generalizations to multilattices [17]see Section C1.

The paper [26] introduces several normality relations and characterizes themby appropriate relative closure operators. For instance, the operator y−x = y∧xis associated with the normality relation

xKy ⇐⇒ x ≥ y and x ∧ y = y .

Then every two finite K-normal chains with common extremities have isomor-phic Zassenhaus refinements.

PART C

C1. Multilattices

In 1953 Mihail Benado introduced the concept of a multilattice in a paper [12]which announces two equivalent definitions of this concept and a few examples.

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The standard reference for the theory of multilattices is the paper [17] (withcorrections in [24]), while other papers of Benado on multilattices and theirapplications are [15], [16], [33], [35], [36]. [39], [40], [51]. In this Section weselect several definitions and theorems from [17].

The basic definition is the following. Let M be a partially ordered set. Ifa, b ∈ M , denote by U(a, b) and L(a, b) the (possibly empty) sets of upperbounds of a, b and of lower bounds of a, b, respectively. The poset M is amultilattice if for any c ∈ U(a, b), the set U(a, b) ∩ x ∈ M | x ≤ c has aminimal element, and dually. Note that lattices are defined by the strongerrequirement that for all a, b ∈ M , the non-empty sets U(a, b) and L(a, b) haveleast element and greatest element, respectively.

It is easily seen that this definition is equivalent to the following one:A multilattice is a poset M such that the following conditions are fulfilled

for any a, b ∈ M:(MLsup) every upper bound of a, b includes a minimal upper bound,and(MLinf) every lower bound of a, b is included in a maximal lower bound;if there is no common upper bound of a, b/common lower bound of a, b, thencondition (MLsup)/condition (MLinf) is vacuously satisfied.

The definition of multilattices can also be given in terms of the follow-ing three well-known pre-orders (which are important in the context of non-deterministic programming languages):

X S Y ⇐⇒ ∀ y ∈ Y ∃x ∈ X x ≤ y (Smyth ordering) ,

X H Y ⇐⇒ ∀x ∈ X ∃ y ∈ Y x ≤ y (Hoare ordering),

X EM Y ⇐⇒ X S Y & X H Y (Egli−Milner ordering) .

A remarkable theorem says that a multilattice M is a lattice if and only ifM is directed, i.e., every two elements a, b ∈ M have upper bounds and lowerbounds, and has the Riesz Interpolation Property (RIP), i.e., if ai, bj ∈M satisfyai ≤ bj (i, j = 1, 2), then there exists c ∈M such that ai ≤ c ≤ bj (i, j = 1, 2).

For every upper bound Ω of a, b, let (a∨b)Ω denote the set of minimal upperbounds of a, b included in Ω, and dually, let (a ∧ b)ω denote the set of maximallower bounds that include the lower bound ω. Then the set-theoretical uniona ∨ b of all (a ∨ b)Ω is the (possibly empty) set of all minimal upper bounds ofa, b and the set-theretical union a∧b of all (a∧b)ω is the (possibly empty) set ofall maximal lower bounds of a, b. Benado proved the following properties (Mn)and their duals (Mn′):(M1) (commutativity) if a ∨ b 6= ∅ then a ∨ b = b ∨ a;(M2) (partial associativity) if a ∨ b 6= ∅ and (a ∨ b) ∨ c 6= ∅, then b ∨ c 6=∅, a ∨ (b ∨ c) 6= ∅ and for every m ∈ (a ∨ b) ∨ c there is m′ ∈ a ∨ (b ∨ c) suchthat m ∨m′ 6= ∅ and m ∨m′ = m;(M3) (absorption) if a ∨ b 6= ∅ then a ∧ (a ∨ b) = a;(M4) a ∨ a 6= ∅;

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(M5) if the elements a, b,m,m′ satisfy m,m′ ∈ a ∨ b, m ∨ m′ 6= ∅ andm 6= m′, then m,m′ 6∈ m ∨m′.

Conversely, if M is a set endowed with two multioperations ∨,∧ : M2 −→P(M) which satisfy (M1)-(M5) and their duals, then a ∨ b = b ⇐⇒ a ∧ b = aand the relation a ≤ b⇐⇒ a∨ b = b⇐⇒ a∧ b = a is a partial order on M , suchthat conditions (ML) above are satisfied and a ∨ b/a ∧ b is the set of minimalupper bounds of a, b/of maximal lower bounds of a, b.

Thus multilattices can also be defined as multialgebras, i.e., as sets endowedwith two multioperations a∨b, a∧b satisfying the above axioms. Jakubık [Jak1],solved two problems raised by Benado in [17], namely he proved that (i) axiom(M5)&(M5′) does not follow from the other axioms of the system above, and(ii) if the multioperations ∨ and ∧ are associative and M is directed, then themultilattice M is in fact a lattice, the latter hypothesis being essential.

Multilattices appear in various contexts, such as divisibility theory [14], [15](see the next Section), Jordan-Dedekind chain conditions, the Mobius func-tion [33], [35], [40], partial differential equations (the paper [16] solves a prob-lem about the wave equation in two-dimensional space-time raised by G. Birk-hoff [1948, Ch.9,13]), functional analysis, theory of geometrical continua (e.g.squares ordered by set inclusion), or topological complexes. See [17] and thesurvey papers [39], [51].

Benado introduces the concepts of modular multilattice and distributive mul-tilattice by generalizing two well-known charcterizations of modular lattices anddistributive lattices, respectively. A multilattice M is distributive (modular) ifthe conditions v ∈ a ∨ b, v ∈ a ∨ b′, u ∈ a ∧ b, u ∈ a ∧ b′ (and b ≥ b′) imply thatb = b′.

Lihova and Repasky [LiRe] have proved that the classes of directed modularmultilattices and directed distributive multilattices are varieties, that is, theyare closed under the constructions of submultialgebras, homomorphic algebrasand direct products.

The set of all squares with the sides parallel to two rectangular directions,partially ordered by set inclusion, is a multilattice which is not a lattice; cf.Benado [17], page 312. This multilattice is modular, since the map v whichassigns to any square a the length v(a) of its side, is a norm, as shown on page336, and any multilattice pssessing a norm is modular by Theorem 5.5 on page337. It would be interesting to check when the multilattices provided by words,partially ordered by ≤pf , ≤sf or ≤fct (see C3), are modular or distributive.

The following reflexive relation is introduced between the quotients x/y (i.e.,intervals [y,x]): x/y ' x′/y′ provided x ≥ x′, y ≥ y′, x ∈ (x′ ∨ y)x andy′ ∈ (x′ ∧ y)y′ . The transitive closure ∼ of ' is called T-similarity. Then thefollowing generalization of the Schreier refinement theorem holds: in a modularmultilattice, two chains with common extremities possess a system of canonicalrefinements having the same length and whose conjugate quotients are T-similar.

A technique introduced by Benado in [14] and largely used throughout his

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entire work is the use of what he calls Dedekind quadrilaterals or simply quadri-laterals. A quadrilateral is a very particular case of two chains with commonextremities, namely a quadruple (Ω, ω; a, b), where Ω ≥ a, b ≥ ω. A subquadri-lateral of (Ω, ω; a, b) is any quadrilateral (m, d; a, b) with m ≤ Ω and ω ≤ d.

Given a quadrilateral (Ω, ω; , b) and two elements m ∈ (a ∨ b)Ω and d ∈(a ∧ b)ω, Benado defines the Dedekind connections

χ(p) = (b ∧ p)d, p ∈ Ω/a and ϕ(p′) = (a ∨ p′)m, p′ ∈ b/ω,

and similarly∼χ (q) and

∼ϕ (q′) by interchanging the roles of a and b. A strong

multilattice is a multilattice M such that any quadrilateral contains a subquadri-lateral for which the corresponding Dedekind connections are functions from Mto M .

The following strengthening of the previous Schreier refinement theoremholds: in a strong modular multilattice, every two chains with common extrem-ities have isomorphic refinements.

A well-known paper of Glivenko [1937] suggested to Benado the idea ofconstructing a theory of valuations for multilattices. He introduced a conceptof normed multilattice and proved that any normed multilattice is modular andbesides, if a normed multilattice is filtered, then it can be made into a metricspace.

One more theorem says that every distributive multilattice is strong.

We note that different generalizations of lattices have been proposed in theliterature, some of them similar to multilattices, which however are not quoted.

A poset is called a hypolattice if each of its closed intervals is a lattice andthe lattice operations coincide on overlapping intervals; cf. Draskovickova et al.[Dra].

Mittas and Konstantinidiou [MitKo] introduced superlattices, which arestructures equipped with two multioperations a ∨ b and a ∧ b verifying someaxioms (commutativity, associativity, particular forms of idempotency and ab-sorption, together with conditions allowing to introduce a partial order like inthe case of lattices). These authors had also introduced in 1977 the concept ofa hypolattice, where only join is a multioperation. Both multilattices and hy-perlattices are particular cases of superlattices. A superlattice is a lattice if andonly if it satisfies the identity (a∨ b)∧ a = (a∧ b)∨ a = a. Strong superlatticeswere investigated by Jakubık [Jak4].

Schroder [Sch] considered truncated lattices, which are structures with par-tially defined lattice operations satifying the axioms of idempotency, commuta-tivity and weak associativity.

A nearlattice is a join semilattice such that every principal filter [a) is alattice with respect to the semilattice order; cf. [ChK1],[ChK2],[ChK3].

We note that while these authors take a purely algebraic approach, i.e., basedon multioperations satisfying convenient axioms, the generalization of Benadois mainly based on the partial order.

C2. Divisibility theory

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One of the themes studied by Mihail Benado was divisibility theory [14],[15], [53].

In [14] he introduces the concept of a division semigroup (in French, semi-groupe a division or semigroupe divisionnaire). By this term is meant a semi-group S such that (i) S has an identity 1 and a zero 0, (ii) S\0 is cancellative,(iii) aS = Sa for any a ∈ S, and (iv) ab = 1 implies a = b = 1.

The name “division semigroup” is justified by the fact that the divisibilityrelation a ≤ b ⇐⇒ ∃c a = bc ⇐⇒ ∃d a = db is a partial order such that0 ≤ a ≤ 1 and a ≤ b =⇒ xay ≤ xby.

The following condition H plays a central role in [14]: a poset satisfies Hprovided there exists an element ω0 such that for any quadrilateral (Ω, ω0; a, b)there is a subquadrilateral (m, d; a, b) such that the order isomorphisms m/a↔b/d and m/b↔ a/d hold.

The following main results are proved in [14]:Corollary 5.83. Every division semigroup satisfies H with ω0 = 0.Theorem 5.9. If two chains of a division semigroup have common extremities,

then they have isomorphic refinements.Theorem 5.93. LetR be an integral domain with identity and P be the lattice

of its principal ideals. Then P satisfies the ascending chain condition (ACC) ifand only if R satisfies the “Satz von der eindeutigen Primfaktorzerlegung”, inwhich case the lattice P is distributive.

In other words, the following properties hold:α) If (P,⊆) is a lattice satisfying the ascending chain condition (ACC), then

R is a unique-factorization domain (UFD).β) Conversely, if R is a UFD, then (P,⊆) is a lattice satisfying ACC.γ) If R is a UFD, then the lattice P is distributive.A very short proof in [14] provides a reference to Birkhoff [1948] and a hint.

We give below a more explicit proof.Recall first that an integral domain is a nontrivial commutative ring R with

no zero divisors. One defines (see e.g. [And]): the divisibility relation x|y ify = dx for some d ∈ R; x is a unit if xy = 1 for some y ∈ R, and let U(R)be the group of units of R; x and y are associates, x ∼ y, if x = uy for someu ∈ U(R); a nonzero nonunit x is irreducible if x = yz implies y ∈ U(R)or z ∈ U(R), and is prime if x|yz implies x|y or x|z. Clearly every prime isirreducible , but the converse does not hold in general. An integral domain R isa unique factorization domain (UFD) if every nonzero nonunit of R is a productof irreducible elements of R, and this decomposition is unique up to the orderof factors and association.

Clearly the set P of principal ideals of an integral domain P , equipped withthe product of ideals (a)(b) = (ab), is a division semigroup with identity (1) = Rand zero (0) = 0. Note that (a) ⊆ (b) ⇐⇒ b|a.

α) It is well known that ACC for P implies that every element of R admitsat least one representation as a product of irreducible elements (see e.g. [And],fourth Theorem), hence it remains to prove the uniqueness of this representation.Since P is a lattice, for any two elements a, b ∈ R there exist (d) = l.u.b.((a), (b))in P . It follows by [Doc], Theorem 2, that d = g.c.d.(a, b) in R. Therefore,

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by Theorem 6 in [Doc], every irreducible of R is prime, and this implies theuniqueness of the decomposition into irreducibles by a well-known argument.

β) A well-known construction from number theory shows that any two ele-ments a, b ∈ R have l.c.m. and g.c.d., hence (a), (b) ∈ P have g.l.b. and l.u.b.So P is a lattice. Besides, if (a1) ⊆ (a2) ⊆ (a3) ⊆ . . . is an increasing sequencein P , then a1, a2, a3, . . . is a decreasing sequence in R such that ak+1|ak for allk, hence all the factors of ak+1 are associates of the factors of ak. Therefore thesets of non-unit factors of a1, a2, a3, . . . form a decreasing sequence, hence thatsequence stops at a certain step n; this implies an = an+1 = . . . .

γ) A well-known construction from number theory shows that for any twoelements a, b of a UFD we have ab = [l.c.m.(a, b)][g.c.d.(a, b)]. If P is a lattice,this yields (a)(b) = [(a) ∧ (b)][(a) ∨ (b)], by [Doc], Theorem 2. Therefore, if(a)∧(b) = (a)∧(b′) and (a)∨(b) = (a)∨(b′), then (ab) = (a)(b) = (a)(b′) = (ab′),showing that ab ∼ ab′, hence b ∼ b′, therefore (b) = (b′). But it is well knwonthat the uniqueness of the relative complements implies the distributivity of thelattice.

In partially ordered groups we have x(x ∧ y)−1y = x ∨ y, for any x, y, theexistence of one side implying the existence of the other side and the equality.

The meaning of the above formula allows us to consider partially definedlattices. Moreover, similar formulas have been obtained for multilattice-orderedgroups; cf. Vaida [Va], [Vai]. A classical model for dealing with partially orderedalgebraic structures is the monograph by Fuchs [Fu], whose principal sourceis the classical article of Birkhoff [Bir] on lattice-ordered groups; for partialalgebras see [LjEv].

In the paper Benado [15] (in Romanian and therefore with a very limitedecho), the author considers the divisibility of the algebraic number field R(θ).The problem raised concerns the possible divisibility rules for these integersrather than those for ideals.

As it is known, Dedekind and afterwards Emmy Noether gave an abstractsetting for the theory of ideals. Noetherian rings were introduced and funda-mental factorization theorems for rings were proved. Later on, a program wascarried on by Ore, Ward and Dilworth, aiming at a lattice-theoretic setting forthe Noetherian theory of ideals, which was obtained in a paper of Dilworthpublished in 1961, in the form of a version of the Krull principal ideal theorem.

The main concern in [15] is the study of multilattice/lattice division semi-groups, meaning those division semigroups (S, ·, 0, 1,≤) for which (S,≤) is amultilattice/lattice.

Benado observes that every Noetherian (i.e., satisfying ACC) division semi-group is a multilattice division semigroup.

After several elementary properties, as for instance a∨ b = ∅ ⇐⇒ a∧ b = ∅,Benado proves ([15], Lemma 3.3) that a multilattice division semigroup S islattice-ordered if and only if for any a, b ∈ S and m ∈ (a ∧ b)0, we have m ≥ aband m ≥ ba.

An auxiliary result is the following. Theorem ([15], Lemma 3.4). In a dis-tributive multilattice semigroup, if (a ∨ b)1 = 1 then ab = ba.

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Sketch of proof. Since aS = Sa, there is b1 ∈ S such as ab = b1a. Benadoproves that (a ∨ b)1 = 1 and besides, for m = ab one has (a ∧ b)m = m by thecomputation rules in S and also (a ∧ b)m = m1 (m1 = ba). We have m1 = mand therefore b1 = b by distributivity.

This proof provides an instructive use of distributivity and suggests possibleways to obtain sufficient conditions for commutativity.

Corollary (3.4 in [15]). A Noetherian distributive multilattice semigroup Sis commutative.

Indeed, if S is Noetherian then there exist maximal elements 6= 1 and irre-ducible in the sense of divisibility, and for any such elements p, q, p 6= q, wehave (p ∨ q)1 = 1 and therefore pq = qp by Lemma 3.4. Since S is Noetherian,every element 6= 0 is a product of irreducible elements.

We note that in [JK] the authors raise the problem of whether the assumptionof distributivity or commutativity of the group G considered in their Theorem4.11 can be dropped. The Corollary quoted above gives a sufficient condition forcommutativity. Moreover, in [JK] one says that a quadruple a, b, u, v is regularif u ∈ a∧b, v ∈ a∨b and v−a = b−u (pag. 603), i.e., a+b = u+v. Consideringthe latter expression written in a convenient form for the noncommutative case,we might hope to arrive at a sufficient condition for the equality of the relativecomplements of a, provided they constitute regular quadruples (by an argumentsimilar to that used for γ above).

The principal result of Benado is the following.Theorem ([15], §4, p.267). Let S be a Noetherian division semigroup. If its

multilattice reduct is distributive, then S is a lattice.In general, the unique factorization theorem is not satisfied by the integers

of a field R(θ) and therefore the multilattice division semigroup defined withrespect to divisibility is not lattice-ordered. It follows that this multilattice isnot necessarily distributive and thus distributivity is not a general rule for thedivisibility of integers in algebraic number fields.

As to the proof of the above Theorem, we should notice the interesting wayin which distributivity is used. For in general, in order to obtain the classicaldecomposition of ideals into a product of irreducible ideals one looks for sufficientconditions to ensure that the implication irreducible =⇒ prime holds for ideals,as shown below. The use of distributivity makes ncessary a quite differentapproach. Indeed, in the conventional approach, if ab ≤ p with p irreducible, ifa 6≤ p then gcd(p, a) = 1 and thus gcd(pb, ab) = b. Since pb, ab ≤ p, we haveb ≤ p, showing that p is prime. The last implication is not true if we have amultilattice instead of a lattice.

Benado [15] starts by showing that if p, q are irreducibele, p 6= q and m ∈(p ∧ q)0, then m = pq = qp ; therefore (p ∧ q)0 is a singleton. In the next stephe proves that if c ∈ (a ∨ b), a, b < c and a 6= b, then again a ∧ b and a ∨ b aresingletons. The hypothesis that S is Noetherian is used only in the final step,which is an induction on the lengths of [a, 1] and [b, 1].

We conjecture that the particular use of distributivity is a promising wayfor other applications to decomposition theorems.

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There are three papers of Benado with the title “Remarks on the theory ofmultilattices”. The series is incomplete: it comprises the papers [30] (withoutseries number), [40] (# IV) and [53] (# VI). Paper [53] has the secondarytitle “Contributions a la theorie des structures algebriques ordonnees”, whichnowadays seems too vague if we refer to the contemporary subject “Orderedstructures” (AMS Classification 06Fxx).

As a matter of fact, this paper originates in the courses given by Dan Bar-bilian in the 1950s on the unique factorization theorem in rings of quadraticintegers7, with references to ideals, following Dedekind and Emmy Noether.The theory presented by Benado in [53], closely related to multilattices andhaving one of its main sources in the lectures of Barbilian, deals with decom-position theorems generalizing those of Lasker-Noether and Kurosh-Ore, andbuilds a general framework for divisibility and residuation.

The reader of [53] may encounter difficulties in going through many defini-tions, whose lattice- or multilattice-theoretical background is not made explicitby Benado. An effort to reveal this background is worth doing, as it may en-lighten a theory of divisibility far more general than the conventional one.

C3. Applications in computer science and mathematicalmodelling

Stricto sensu, there are no results in the work of M. Benado which mightbe seen as properly related to computer science. However some of his papersare now quoted as an adequate framework for the algebraic basis of theoreticalcomputer science. In the following examples, references and our own notes aregiven in this sense.

In an earlier stage of computer science, many computer scientists found amajor interest in the lattice-theoretical approach, in Scott’s theory of continuouslattices and in the ADJ work dealing with ω-complete sets and ω-continuousalgebras. However, many structures of interest in computer science are notactually lattices, thus requiring a more general theory of order structures.

A Basic Example. Let Σ∗ be the set of all strings, or words, including theempty string e, over a finite alphabet Σ, and let u ≤pf v iff v = uv1, u beingcalled a prefix of v. Similarly, one defines the dual u ≤sf v iff u is a sufix of v,i.e., v = v2u, and u ≤fct v iff u is a factor of v, i.e., v = v1uv2(e.g., roma ≤pf roman ≤pf romania, man ≤sf roman, man ≤fct romania).Obviously, the relations≤h, for h = pf, sf, fct, are partial orders. In general,twodifferent words u and v may not have a pf - or sf - upper bound or lower bounddifferent from e (e.g. u = ab and v = ba, where a, b ∈ Σ), but they always havethe fct-upper bounds u1vu2uu3 and u1uu2vu3 and possibly others (e.g., ab andba have the upper bounds aba and bab).

7The present authors were among the students of Barbilian in 1956-1957, when he gave acourse on algebraic number theory and a course on the foundations of geometry. The secondauthor attended as well Barbilian’s lectures on the multiplicative theory of ideals, as theunique student (!).

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Remark. If u ≤h v then |u| ≤ |v|, where |w| is the length of w, hence (Σ∗,≤h)is locallyfinite, in the sense that every interval is finite, for h = pf, sf, fct.Moreover, L = (Σ∗, ·, e,≤h) is a monoid and a multilattice, the identity e beingthe first element.

Every locally finite poset is also chain-finite and therefore L satisfies boththe ascending and descending chain conditions, i.e., L is an archimedean posetin the terminology of Benado [14].

The one-sided compatibility of the concatenation with the partial orders≤h, h = pf, sf , is readily checked. For the relevance of these partial orders seeLothaire [Lot].

Partially additive semantics. The idea of partially additive semantics is thefollowing. Given a program prg ∈ Prog and its set of data D = Input∪Output,a possible execution execi, i ∈ I, is viewed as a partially defined function

fi = execi(prg) : D −→ D, dom(fi) = di ⊆ Input .

In the model introduced by Manes and Arbib [MA], different executions fi ofprg should have disjoint domains. The semantics is defined as a sum of thepossible executions

sem(prg) = f =∑

(fi | i ∈ I) ∈ P = Pfn(D,D) ,

where Pfn(D,D) is the set of partially defined functions D −→ D, the sumbeing defined in a partially defined monoid, by

dom(f) =⋃

(dom(fi) | i ∈ I), dom(fj) ∩ dom(fk) = ∅, j 6= k ,

f(x) ::= if (∃ i ∈ I) (x ∈ dom(fi)) then fi(x), else undefined .

The reference structure is a partially additive naturally ordered or sum-orderedassociative semiring (Pfn(D,D),

∑, ·, 0, 1,≤) (see Rudeanu and Vaida [RV],

Vaida [Vai1], [Vai2] and further partial semirings). The multiplication is con-catenation, a posible interpretation being the composition of functions, corre-sponding to the sequential execution, dom(0) = ∅, 1 = 1D and f ≤ g meansthat g is an extension of f (the approximation ordering). P is a (additive)-cancellative, i.e., f + f1 = f + f2 implies f1 = f2.

The above structure of Pfn(D,D) is further explained.Definition. A naturally ordered partial semiring, also called a sum-ordered

partial semiring, is a partial algebra (S,+, ·, 0, 1,≤) with the following prop-erties: (S,+, 0) is a partial commutative monoid with a neutral element calledzero, the addition + being partially defined; this monoid is partially associative,in the sense that if one side of the associative rule exists then the other exists aswell and they are equal; (S,+,≤) is partially ordered by a ≤ b iff ∃ c ∈ S b = a+c(in Pfn(D,D) this is equivalent to the approximation ordering previously de-fined); this order is compatible with addition, meaning that if a ≤ b then theexistence of b+d implies that a+d exists as well and a+d ≤ b+d; (S, ·, 1, 0,≤)

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is a partially ordered monoid with identity element 1 and annihilator element 0;the following rules of partial distributivity are satisfied: if b+ c ∈ S exists then

a(b+ c) = ab+ ac and (b+ c)a = ba+ ca ∀ a ∈ S ,

with the meaning that the sums of the right sides exist as well and the equalitieshold.

The structure P = (Pfn(D,D), 0,≤) is not a lattice; as a matter of fact itis not even directed, because f, g ≤ h is possible only if f(x) = h(x) = g(x) forany x ∈ dom(f) ∩ dom(g). Yet it is a meet semilattice with first element 0 bytaking

dom(f ∧ g) = x ∈ D | x ∈ dom(f) ∩ dom(g) and f(x) = g(x) ,

inf(f, g) = f ∧ g with (f ∧ g)(x) = f(x) = g(x) ∀x ∈ dom(f ∧ g) .Consequently, one has the Riesz interpolation property (RIP): if x1, x2 ≤ y1, y2then ∃ z ∈ P such that x1, x2 ≤ z ≤ y1, y2.

Moreover, P is a multilattice. It remains to show that if for f and g thereis h ∈ Pfn(D,D) such that f, g ≤ h, then there exists sup(f, g) = f ∨ g ∈Pfn(D,D). Indeed, we take

dom(f ∨ g) = dom(f) ∪ dom(g) ,

(f ∨ g)(x) ::= if x ∈ dom(f) then f(x) else g(x) ∀x ∈ dom(f ∨ g) .Readily, dom(f), dom(g) ⊆ dom(f ∨ g) and f, g ≤ f ∨ g. From f, g ≤ h weobtain dom(f ∨ g) ⊆ dom(h) and for x ∈ dom(f ∨ g), if x ∈ dom(f) then(f ∨ g)(x) = f(x) since f ≤ h and similarly for g.

In fact P is a particular case of a multilattice, since it is a dual nearlattice.The term nearlattice was introduced by Cornish and Noor [NC] for a conceptpreviously studied by Sholander under a different name. It designates a joinsemilattice in which every principal filter is a lattice with respect to the inducedorder. Alternatively, a nearlattice was described by Chajda and Kolarık [ChK1,ChK2, ChK3] as an algebra with a ternary operation satisfying eight simpleidentities. Hence the class of nearlattices is a variety.

The multilattice join f ∨ g can be seen as a partially defined algebraic sumif we extend the definition of summability from families of disjoint functions tofamilies with overlapping functions, i.e., functions which coincide on the overlapsof their domains. The support structure would be a partially additive naturallyordered associative semiring P1 = (Pfn(D,D),

∑1, ·, 0, 1,≤). In P1 if the sum

f +1 g exists then it coincides with f ∨ g. This sum exists iff f and g arecompatible, in the sense that there exists h ∈ Pfn(D,D) such that f, g ≤ h.

Three more properties of Pfn(D,D) related to its structure can be providedmerely for motivating a further study.The proofs should remain for anotherpaper.

The sum decomposition property (SDP), saying that

0 ≤ x ≤ a+ b =⇒ (∃ s, t) 0 ≤ s ≤ a&0 ≤ t ≤ b&x = s+ t ,

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is equivalent to (RIP) under rather general conditions, but this equivalence isnor proved for P and therefore an independent proof is needed.

Property 1. If f, g, h ∈ Pfn(D,D) and f ≤ g +1 h then g and h exist suchthat g1 ≤ g, h1 ≤ h and f = g1 +1 h ∀x ∈ dom(f).

Property 2. Every directed family of functions has a sup.Property 3. In P1 the relative complements, whenever they exist, are unique.

As a matter of fact, it turns out that mulilattices and several more generalstructures have applications in theoretical computer science and in logic. Weplan to come back in a future paper on the literature which quotes Benado inthis respect. Right now we illustrate the intended applications only by a fewreferences which are not coming from a rigorous selection, being rather a resultof our limited knowledge.

Thus the papers [Cab], [Cord] and [Mart] are examples of the systematic andconstant attention devoted to multilattices by the Spanish school. The lecture[Mark] states that lattices are not enough for the specific needs of computerscience. Moreover, the basic examples given therein concern syntax – wordordering – and semantics – partially defined functions – yet without a specificdevelopment for these examples. The book [DP] does not quote Benado, but theinteresting and new examples of order structures proposed to the reader suggestpossible links for a future investigation, e.g., difference posets or interpolationconditions. The study [Maru] motivates an interest for many-valued logics basedon lattice-ordered algebraic structures and thus possibly based on multilattice-ordered monoids.

Part D

D1. The general theory of partially ordered sets

Benado aimed at a general theory of partially ordered sets, which he cre-ated under this very name. The results were announced in [34], [38], [42]-[46],[49], [50], and full proofs were given in [47], [48], [52]. As curious as it mayseem nowadays, Benado regarded his theory of multilattices as being just anintermediate step towards the general theory of partially ordered sets.

Roughly speaking, the idea is to endow a partially ordered set (poset) witha supplementary structure in such a way that the most important classes ofposets and lattices be obtained as specializations of this general structure.

Technically speaking, Benado considers a poset (P,≤) endowed with twocorrespondences Υ,Σ between P and the set of subsets of P , subject to theconditions

dΥA =⇒ d is an u.b. of A and mΣB =⇒ m is a l.b. of B,where u.b. and l.b. stand for upper bound and lower bound, respectively;besides, it is assumed that tuples (a, b, d,m) such that dΥa, b and mΣa, bdo exist. We refer to all this as a (Υ,Σ) structure.

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A few particular (Υ,Σ) structures play an important role:1) the Dedekind structure (∆0,M0), defined byd∆0A⇐⇒ d is the l.u.b. of A and mM0B ⇐⇒ m is the g.l.b. of B;2) the Hausdorff structure (∆H ,MH), defined byd∆HA⇐⇒ d is a minimal u.b. of A, andmMHB ⇐⇒ m is a maximal l.b. of B;3) the Riesz structure (∆R,MR), defined byd∆RA⇐⇒ d is an u.b. of A and for any u.b. u of A there is an u.b. t of A

such that t ≤ d, u, andmMRB ⇐⇒ m is a l.b. of B and for any l.b. v of B there is a l.b. w of B

such that w ≥ m, v;4) the filtering structure (structure filtrante) (∆F ,MF ), defined byd∆FA⇐⇒ d is an u.b. of A and for every u.b. u of A there is an u.b. t of

A such that t ≥ d, u, anddMFB ⇐⇒ m is a l.b. of B and for every l.b. v of B there is a l.b. w of B

such that w ≤ m, v;5) the discrete structure (∆,M) defined byd∆A⇐⇒ d is an u.b. of A, and mMB ⇐⇒ m is a l.b. of B.Having in view a triple (a, a, a, a), we see that the discrete structure is a

paraphrase of the concept of poset itself. A multilattice is a Hausdorff structuresatisfying the following conditions:

u ≥ a, b =⇒ ∃ d u ≥ d & d∆Ha, b,v ≤ a, b =⇒ ∃m v ≤ m & mMHa, b.The general theory of posets consists in the study of numerous conditions

that a (Υ,Σ) structure may fulfil, meaning the discovery of relationships thatexist between these conditions, called elementary incidence properties. Hereare a few samples. A (Υ,Σ) structure is said to be: natural, if a ≥ b =⇒aΥa, b & bΣa, b; saturated, if d, d′Υa, b&m,m′Σa, b&d ≤ d′ &m ≤m′ =⇒ d = d′ &m = m′; with similarity (a similitude), if conditions dΥa, band mΣa, b imply d = a ⇐⇒ b = m; relatively complemented, if u ≥ a ≥v =⇒ ∃ b uΥa, b& vΣa, b; complemented, if the poset P has 0 and 1 and∀ a∃ b 1Υa, b&0Σa, b; refining (rafinante), if u ≥ a ≥ v =⇒ ∃ d,m u ≥d& v ≤ m& dΥa, b&mΣa, b. Other elementary incidence properties havelonger definitions.

The study of relationships between elementary incidence properties was ex-tended to include several generalizations of the concepts of modular and dis-tributive lattices. For instance, a (Υ,Σ) structure is called:

K-modular, if dΥa, b& dΥa, b′&mΣa, b&mΣa, b′& b ≥ b′ =⇒ b =b′;

O-modular, if the conditions dΥa, b&mΣa, b& d ≥ a′ ≥ a& b ≥ b′ ≥ mimply a′Υa, b′ ⇐⇒ b′Σa′, b;

W-modular, if dΥa, b&mΣa, b =⇒ [a, d] ≈ [m, b] (lattice isomorphism);G-distributive, if dΥa, b& dΥa, b′&mΣa, b&Σa, b′ =⇒ b = b′;

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G′-distributive, if the hypothesis of G-distributivity and d ≥ d′ and m ≥ m′

imply b ≥ b′.The above types of modularity are not equivalent, nor are the variants of

distributivity. Also, distributivity does not imply modularity, unless certainsupplementary hypotheses are assumed.

Among the numerous theorems obtained by Benado in this field we mentiona Schreier-like refinement theorem for W -modular refining structures, and ageneralization of the Glivenko theorem which makes every normed lattice intoa metric space. Another theorem characterizes the Dedekind structure of aBoolean algebra viewed as a poset.

Benado acknowledges the valuable observations made by M. Kolibiar and J.Jakubık in their correspondence with him during the preparation of the generaltheory of partially ordered sets.

D2. Monotone connections

Another research project of Benado concerns the monotone conenctions in-troduced in [5]. Certain peculiarities of the Zassenhaus refinements of α-normalchains and β1-normal chains have led thim to the introduction of the conceptsof monotone connections of type I and II, respectively, while the monotone con-nections of type III have been suggested by the well-known Galois connections.Let P and Q be two posets, and let (+,−) be a pair of isotone (also calledincreasing) functions + : P −→ Q and − : Q −→ P . Then (+,−) is called amonotone connction of type I or II or III between P and Q according as

x+− ≤ x ∀x ∈ P & y−+ ≤ y ∀ y ∈ Q or

x ≤ x+− ∀x ∈ P & y ≤ y−+ ∀ y ∈ Q or

x+− ≤ x ∀x ∈ P & y ≤ y−+ ∀ y ∈ Q .

Recall also that a covariant or isotone Galois connection between P and Q is apair (+,−) of isotone functions + : P −→ Q and − : Q −→ P such that

x ≤ x+− ∀x ∈ P & y−+ ≤ y ∀ y ∈ Q ,

while a contravariant or antitone Galois connection between P and Q is a pair(+,−) of antitone (also called decreasing) functions + : P −→ Q and − : Q −→P such that

x ≤ x+− ∀x ∈ P & y ≤ y−+ ∀ y ∈ Q .

The paper [59] continues the study of monotone connections of type I. Inthe introduction to [59] Benado states that he has introduced the monotoneconnections of type I “rather in order to exhaust the types of monotone con-nections (plutot en exhaustion des especes de connexions monotones), amongwhich those of type III, introduced by J.R. Buchi under the name Paarungen,have been independently rediscovered by myself and by R. Croisot”.

However the aforementioned five concepts do not exhaust the eight possibil-ities of combining “+,− isotone” or “+,− antitone” with “∀x ∈ P x ≤ x+− ”

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or “∀x ∈ P x+− ≤ x ” and with “∀ y ∈ Q y ≤ y−+ ” or “∀ y ∈ Q y−+ ≤ y ”.We want to note here that exhaustion is however realized, to the effect that thestudy of these eight concepts reduces to the study of Benado’s monotone connec-tions of type I and of type III. Indeed, denote by Xop the dual of a poset X. Itwas remarked by Benado himself that the monotone connections of type I andII are dual to each other, to the effect that if + : P −→ Q and − : Q −→ P forma connection of type I (of type II), then + : P op −→ Qop and − : Qop −→ P op

form a connection of type II (of type I). Therefore the properties of monotoneconnections of type II are obtained by duality from the properties of monotoneconnections of type I. It is also immediately seen that the monotone connectionsof type III and the covariant Galois connections are dual to each other, thereforethe study of the latter reduce to the study of the former. We have thus provedthat the study of the four isotone connections (meaning that +,− are isotone)reduces to the study of the monotone connections of types I and III, and itremains to show that the same reduction is valid for the four antitone connec-tions (meaning that +,− are antitone). This follows immediately by observingthat if we associate with each monotone connection + : P −→ Q, − : Q −→ Pof type I or III the two antitone connections + : P op −→ Q, :Q −→ P op and+ : P −→ Qop, − : Qop −→ P , then we obtain the four antitone connections.

The paper [59] is devoted to monotone connections of type I. It studiesinvolutive connections, meaning that ∀x ∈ P x+−+ = x+ and ∀ y ∈ Q y−+− =y−. It also provides characterizations of semilattices and semimultilattices interms of monotone connections. Here is the former theorem.

It is easy to see that if (S,∧) is a meet semilattice, then for every a, b ∈ Sthe mappings

(S0) + : (a] −→ (b], x+ = b ∧ x and − : (b] −→ (a], y− = a ∧ y ,

form a monotone connection of type I such that for every x ∈ (a] and everyy ∈ (b] the following properties hold:

(S1) x+ ≤ x and y− ≤ y ,

(S2) x+− = x⇐⇒ x ≤ a&x ≤ b and y−+ = y ⇐⇒ y ≤ a& y ≤ b .

Conversely, if P is a poset such that for every a, b ∈ P there is a monotoneconnection of type I + : (a] −→ (b], − : (b] −→ (a] satisfying (S1) and (S2),then P is a meet semilattice.

To prove the second statement, take a, b ∈ P and use the monotone con-nection associated with these elements. Then a+ ∈ (b], that is a+ ≤ b, whilea+ ≤ a by (S1). Now take an arbitrary lower bound v of a and b. Then v ∈ (a]and v+− ≤ v (type I). On the other hand v+− = v by (S2) and v ≤ a impliesv+− ≤ a+− ≤ a+ by (S1), hence v ≤ a+. 2.

Benado intended to continue this study by obtaining characterizations ofother ordered structures, such as lattices and multilattices, or Kurepa’s ramifiedsets and trees. Unfortunately, this research project has been interrupted, thepaper [59] being apparently the last paper of Benado.

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D3. Metric lattices

The paper [8], whose results were announced in [4], studies lattices endowedwith a metric. While K. Menger, V. Glivenko and other mathematicians in-vestigate lattices equipped with a valuation which plays the role of a norm inanalysis and which in particular induces a distance, Benado takes the oppositeway. He defines a distance-like function which he studies in great detail; inparticular this function induces a valuation of the intervals of the lattice.

Following M. Frechet, Benado defines a semidistance on a lattice L as afunction d : L3 −→ R satisfying, for all x, y, z ∈ L,

d(x, y) ≥ 0 ,

d(x, y) = 0 ⇐⇒ x = y ,

d(x, y) = d(y, x) ;

if, moreover, d fulfils the triangle inequality

d(x, z) ≤ d(x, y) + d(y, z) ,

then d is called a distance or a metric. We refer to a lattice equipped with asemidistance or a distance as an sd-lattice or a d-lattice, respectively. 8 Thusevery d-lattice is a metric space. A semidistance d on L and the lattice Litself are called regular or convex or concave according as the identity d(x, y) =d(x∨ y, x∧ y) or d(x, y) ≤ d(x∨ y, x∧ y) or d(x, y) ≥ d(x∨ y, x∧ y) holds; notethat a semidistance may have none of these properties. The semidistance andthe lattice are said to be additive provided

x ≥ y ≥ z =⇒ d(x, z) = d(x, y) + d(y, z) .

Taking x := x ∨ y and z := x ∧ y, we see that every additive sd-lattice satisfiesthe weak additivity property 9

d(x ∨ y, x ∧ y) = d(x ∨ y, x) + d(x, x ∧ y) = d(x ∨ y, y) + d(y, x ∧ y) .

The functionsH1(x, y) = d(x ∨ y, x)− d(y, x ∧ y) ,H2(x, y) = d(x ∨ y, y)− d(x, x ∧ y) ,

called torsions, are essentially due to D. Barbilian [1964]. Note that H2(x, y) =H1(y, x). Let us say that an sd-lattice is torsion free if H1 = 0 identically,or equivalently, H2 = 0 identically. This amounts to the equivalent identitiesd(x ∨ y, x) = d(y, x ∧ y and d(x ∨ y, y) = d(x, x ∧ y).

Benado studies the relationships between the above and other properties.Here are a few typical results.

If an sd-lattice satisfies8Benado uses the terms metrisable lattice and metric lattice, respectively. Yet the estab-

lished terms metrisable topological space and metric lattice have different meanings.9Called elementary additivity by Benado.

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(0) x ∧ y ≤ a ≤ x ∨ y =⇒ d(a, x) + d(a, y) = d(a, x ∨ y) + d(a, x ∧ y) ,

then it is regular, torsion free, weakly additive and distributive.To prove this we write down the hypothesis on a, x, y taking in turn a :=

x, a := y, a := x ∨ y, a := x ∧ y :

(1) d(x, y) = d(x, x ∨ y) + d(x, x ∧ y) ,

(2) d(y, x) = d(y, x ∨ y) + d(y, x ∧ y) ,

(3) d(x ∨ y, x) + d(x ∨ y, y) = d(x ∨ y, x ∧ y) ,

(4) d(x ∧ y, x) + d(x ∧ y, y) = d(x ∧ y, x ∨ y) .

The sums (1) + (2) and (3) + (4) yield

2d(x, y) = d(x∨ y, x) + d(x∨ y, y) + d(x, x∧ y) + d(y, x∧ y) = 2d(x∨ y, x∧ y) ,

hence the lattice is regular. Now (1) together with the regularity condition and(4) yield

d(x, x ∨ y) + d(x, x ∧ y) = d(x, y) = d(x ∨ y, x ∧ y) = d(x, x ∧ y) + d(y, x ∧ y) ,

hence d(x, x∨y) = d(y, x∧y), that is, the lattice is torsion free. This transforms(3) into d(y, x∧y)+d(x∨y, y) = d(x∨y, x∧y), showing that d is weakly additive.

It is well known that a lattice is distributive if and only if it satisfies

a ∨ x = a ∨ y & a ∧ x = a ∧ y =⇒ x = y .

So we assume the elements a, x, y fulfil a∨x = a∨ y and a∧x = a∧ y and mustprove that x = y. But these elements satify a ∧ y ≤ x ≤ a ∨ y, so that takinga := x and x := a in the hypothesis (0) we get

d(x, a) + d(x, y) = d(x, a ∨ y) + d(x, a ∧ y) .

Using again the hypothesis (0) for a ∧ x ≤ x ≤ a ∨ x and the hypothesis ona, x, y, we obtain

d(x, a) = d(x, a ∨ x) + d(x, a ∧ x) = d(x, a ∨ y) + d(x, a ∧ y) .

By comparing the results we obtain d(x, a) + d(x, y) = d(x, a). So d(x, y) = 0,therefore x = y, completing the proof.

Any additive distance is convex.For the hypotheses imply

d(x ∨ y, x ∧ y) = d(x ∨ y, x) + d(x, x ∧ y) ,

d(x ∨ y, x ∧ y) = d(x ∨ y, y) + d(y, x ∧ y) ,

d(x, y) ≤ d(x, x ∨ y) + d(x ∨ y, y) ,

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d(x, y) ≤ d(x, x ∧ y) + d(x ∧ y, y) ,

hence 2d(x, y) ≤ 2d(x ∨ y, x ∧ y), as was claimed.Any additive regular distance is torsion free.For the hypotheses imply

d(x ∨ y, y) + d(y, x ∧ y) = d(x ∨ y, x ∧ y) = d(x, y) ≤ d(x, x ∨ y) + d(x ∨ y, y) ,

d(x ∨ y, x) + d(x, x ∧ y) = d(x ∨ y, x ∧ y) = d(x, y) ≤ d(x, x ∧ y) + d(x ∧ y, y) ,

therefore d(y, x∧y) ≤ d(x, x∨y) and d(x∨y, x) ≤ d(x∧y, y), that is, H1(x, y) =0.

The following converse holds: any torsion free additive regular semidistanceis a distance. Therefore an additive regular semidistance lattice is a distancelattice if and only if it is torsion free. A large part of the paper is devoted tothese lattices, called elementary d-lattices. Other classes of sd-lattices are alsointroduced and studied, many results being expressed in terms of valuation. Thismeans that if x ≤ y, then d(x, y) is interpreted as a valuation of the interval[x, y] = z ∈ L | x ≤ z ≤ y and this is written in the form

d(x, y) = v(y

x) .

The research, suggested by previous works of Glivenko [1936] and Barbilian[1946], generalizes several results of these authors.

Benado has gone farther in [17], where he introduced the concept of a normal-ized multilattice (multistructure normee). This means a multilatticeM equippedwith a map v : M −→M (valuation) which satisfies the conditions

a < b =⇒ v(a) < v(b) ,

a ∨ b 6= ∅ 6= a ∧ b =⇒ v(a) + v(b) = v(m) + v(d) (∀m ∈ a ∨ b) (∀ d ∈ a ∧ b) .

The following theorems generalize well-known results of Glivenko for lattices:1. Every normalized multilattice is modular.2. Every directed normalized multilattice is a metric space with respect to

the distance ρ(a, b) = v(m)− v(d).Another approach was taken by Kolibiar [Kol1], who defined a metric mul-

tilattice as a multilattice which is also a metric space with respect to a distanceρ which satisfies the following supplementary conditions:

a ≤ b ≤ c =⇒ abc, x ∈ a ∨ b =⇒ axb, x ∈ a ∧ b =⇒ axb ,

where xyz denotes metric betweenness, i.e., ρ(x, y) + ρ(y, z) = ρ(x, z). Thedirected metric multilattices coincide with Benado’s normalized multilattices.

Two metric lattices M and M ′ are said to be m-equivalent if there is a bijec-tion ϕ : M −→ M ′ such that abc ⇐⇒ ϕ(a)ϕ(b)ϕ(c). In [Kol1], m-equivalenceis studied in some detail. Sample result: two directed distributive metric mul-tilattices M and M ′ are m-equivalent if and only if there are two multilattices

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A1 and A2 such that M ∼ A1 × A2 and M ′ ∼ A1 × A2, where ∼ denotes iso-morphism and A is the dual of A. In [Kol2], necessary and sufficient conditionsare given for a metric space to be partially orderable so as to become a metricmultilattice.

A related concept was introduced by Jakubık [Jak4]. A graph isomorphismbetween two multilattices M,M ′ is a bijection ϕ : M −→ M ′ such that twoelements a, b ∈ M are neighbouring (benachtbar) if and only if ϕ(a), ϕ(b) areneighbouring. The meaning of this term is that a ≺ b or b ≺ a, where x ≺ ymeans that x < y and there is no z satisfying x < z < y. Besides, a multilatticeM is said to be of finite length (langenendlich) if every bounded chian in M isfinite. Jakubık has proved a theorem which can be paraphrased as follows: twodirected distributive multilattices of finite length are graph-isomorphic if andonly if they are m-equivalent.

D4. Diametric spaces

The theory of diametric spaces is the last field initiated and explored byBenado. The papers devoted to this subject are [54]-[58] and a few manuscriptswhich are announced in the references of the printed papers and which appar-ently have remained unpublished.

A diametric space is a set E endowed with a function δ : E×E −→ R whichsatisfies the conditions

(E1) δ(a, b) = δ(b, a) ,

(E2) δ(a, b) + δ(c, c) ≤ δ(a, c) + δ(b, c) .

The elements of E are called bodies (Korper), while δ(a, b) represents the spatialdimension (Maßraum) or diameter of a and b.

Several examples of diametric spaes are listed in [54], but unfortunately thereader is referred to an unpublished manuscript for details. However the reviewMR38#3774 to [55] makes explicit the following diametric space. Let E be theset of all spheres of a metric space and take δ(a, b) = θ(a, b)+ r(a)+ r(b), whereθ(a, b) is the distance between the centers of a and b, while r(c) is the radius ofc. Having this example in mind, the motivation of most of Benado’s definitionsbecomes clear.

In every diametric space E the following function θ is also introduced:

(E3) 2θ(a, b) = 2δ(a, b)− δ(a, a)− δ(b, b) .

It is easy to see that θ is a quasimetric, that is, θ(a, a) = 0, θ(a, b) = θ(b, a)and θ(a, b) ≤ θ(a, c) + θ(c, b). Conversely, if θ is a quasimetric on a set E andχ : E −→ R, then E becomes a diametric space with

(E4) δ(a, b) = θ(a, b) + χ(a) + χ(b) .

Having in view that θ(a, b) = 0 & δ(a, a) = δ(b, b) ⇐⇒ δ(a, b) = δ(a, a) =δ(b, b), the following equivalence relation on E is taken in the role of equalityon E:

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(E5) a = b⇐⇒ δ(a, b) = δ(a, a) = δ(b, b) .

To prove the transitivity of this relation, suppose

δ(a, b) = δ(a, a) = δ(b, b) = δ(c, c) = δ(b, c) .

Then δ(a, b) + δ(c, c) ≤ δ(a, c) + δ(c, b) implies δ(c, c) ≤ δ(a, c), while δ(a, c) +δ(b, b) ≤ δ(a, b) + δ(b, c) implies δ(a, c) ≤ δ(a, b) = δ(c, c). Therefore δ(a, c) =δ(c, c) and we have also δ(a, a) = δ(c, c).

The following property is important:

(E6) δ(b, a) ≤ δ(a, a) =⇒ δ(b, c) ≤ δ(a, c) .

Indeed, from δ(b, a) ≤ δ(a, a) we infer

δ(b, c) + δ(b, a) ≤ δ(b, c) + δ(a, a) ≤ δ(b, a) + δ(a, c) ,

hence δ(b, c) ≤ δ(a, c).Having in view that θ(a, b) + 1

2δ(b, b) ≤12δ(a, a) ⇐⇒ δ(a, b) ≤ δ(a, a), the

relation “b is interior to A”, written b ≤ a, is defined by

(E7) b ≤ a⇐⇒ δ(b, a) ≤ δ(a, a) .

This relation is a partial order on E. Indeed, reflexivity is trivial, transitivityfollows immediately by (E6), and if δ(b, a) ≤ δ(a, a) and δ(a, b) ≤ δ(b, b), thenδ(b, b) ≤ δ(a, b) and δ(a, a) ≤ δ(b, a), therefore δ(b, a) = δ(a, a) and δ(a, b) =δ(b, b), hence a = b. The order (E6) is called the intrinsic or natural order of(E, δ).

The relative positions “b is internally tangent to a”, “a and b are concentric”,and “c is between a and b”, are defined by b IT a ⇐⇒ δ(b, a) = δ(a, a), a CCb⇐⇒ θ(a, b) = 0 and c β ab⇐⇒ θ(a, b) = θ(a, c) + θ(c, b), respectively.

Besides the above concepts there are many other operations and relationsintroduced on a diametric space and whose relationships have been studied ingreat detail. Morphisms between diametric spaces have been also introduced.If (E, δ) and E′, δ′) are diametric spaces, a map ϕ : E −→ E′ is said to be adiametric isomorphism if the equalty δ(a, b) = δ′(ϕa, ϕb) holds for every a, b ∈E; this implies that the associated posets E,≤) and (E′,≤′) are isomorphic.There are several variants of this concept, studied in some detail.

Another favourite research theme refers to interpolation properties. To statemany of these properties one needs the following notation. If ρ is a binaryrelation on E and A,B ⊆ E, then we define AρB ⇐⇒ (∀ a ∈ A) (∀ b ∈ B) aρb;as usual, we write aρB and Aρb instead of aρB and Aρb. The interpolationproperties are certain implications which a diametric space may or may notsatisfy. For instance, A ≤ B =⇒ ∃ c A ≤ c ≤ B and A IT B =⇒ ∃ c A ≤ c ≤ B,where ≤ is the natural order and IT is the relation “internally tangent”, aretwo interpolation properties. The relationships between many interpolationproperties have been studied, with the aim of establishing sufficient conditionsfor their validity.

30

Another concern of Benado was the construction of diametric spaces fromgiven partially ordered sets. Here is a sample theorem. Recall that the conceptof a ramified set, introduced and studied by D. Kurepa, means any poset whoseorder ideals are chains. Suppose (K,≤) is a dually directed ramified set forwhich there is a strictly increasing function f : K −→ R. Then K can bemade into a diametric space (K, δ) which satisfies δ(a, b) = max(δ(a, a), δ(b, b))whenever a and b are comparable. Besides, the natural order of (K, δ) is finerthan the original order of K, i.e., a ≤ b =⇒ δ(a, b) ≤ δ(b, b).

Denote by H the hypothesis of the above theorem. A certain condition Chas been found such that H&C =⇒≤ coincides with the natural order=⇒ C.

Part E

Papers of Mihail Benado

1. Series canoniques principales du probleme de Galois. Bull. Math. Soc.Roumaine Sci. 47(1946), 49-61.

2. Nouveaux theoremes de decomposition et d’intercalation attaches a lanormalite α. C.R. Acad. Sci. Paris 228(1949), 529-531. MR 10, p.502.

3. Le fondement axiomatique du theoreme Jordan-Holder relatif aux seriesprincipales. C.R. Acad. Sci. Paris 229(1949), 332-334. MR 11, p.309.

4. Teoria structurilor metrizabile. Bul. Sti. Acad. R.P. Romane A1(1949),353-359. MR 12, p.237.

5. Notiunea de normalitate si teoremele de descompunere ale algebrei. Stud.Cerc. Mat. 1(1950), 282-317. MR 16, p.212.

6. Normalitatea α si teorema Jordan-Holder. Bul. Sti. Ser. Mat. Fiz.Chim. 2(1950), 557-560.

7. Normalitatea Uzkov si teorema Jordan-Holder. Com. Acad. R.P.Romane1(1951), 7-11. MR 17, p.7.

8. Teoria structurilor metrizabile. Stud. Cerc. Mat. 2(1951), 45-106;erratum 285. MR 16, 275i.

9. Asupra teoremei de rafinare a lui O. Schreier. Com. Acad. R.P. Romane1(1951), 1021-1023. MR 17, p.7.

10. Asupra teoremelor de descompunere ale algebrei. Stud. Cerc. Mat.3(1952), 263-288. MR 16, p.212.

11. Multimile partial ordonate si teorema de rafinare a lui Schreier. Bul.Sti. Sect. Sti. Mat. Fiz. 4(1952), 585-591. MR 15, p.595.

12. Asupra unei generalizari a notiunii de structura. Bul. Sti. Sect. Sti.Mat. Fiz. 5(1953), 41-48. MR 16, p.668.

13. Teoria abstracta a relatiilor de normalitate. Stud. Cerc. Mat. 4(1953),69-120. MR 16, p.212.

14. Les ensembles partiellement ordonnes et le theoreme de raffinement deSchreier. I. Czechoslovak Math. J. 4(79)(1954), 105-129. MR 16, p.668.

15. Asupra teoriei divizibilitatii. Bul. Sti. Sect. Sti. Mat. Fiz. 6(1954),263-270. MR 16, p.668.

31

16. Asupra unei probleme a lui Garrett Birkhoff. Bul. Sti. Sect. Sti. Mat.Fiz. 6(1954), 703-739. MR 17, p.341.

17. Les ensembles partiellement ordonnes et le theoreme de raffinement deSchreier. II. Czechoslovak Math. J. 5(80)(1955), 308-344. MR 17, p.937.

18. Asupra descompunerii unui grup ın produs direct. I. Bul. Sti. Sect. Sti.Mat. Fiz. 7(1955), 241-248. MR 17, p.1050.

19. Asupra descompunerii unui grup ın produs direct. II. Bul. Sti. Sect.Sti. Mat. Fiz. 7(1955), 249-254. MR 17, p.1050.

20. Asupra unei probleme din teoria conditiilor de normalitate Ore. Com.Acad. R.P. Romane 5(1955), 1241-1243. MR 17, p.937.

21. Uber die allgemeine Theorie der regularen Produkte von Herrn O.N.Golovin. I. Math. Nachr. 14(1955), 213-234 (1956). MR 18, p.871.

22. Asupra descompunerii unui grup ın produs direct. III. Bul. Sti. Sect.Sti. Mat. Fiz. 8(1956), 5-10. MR 19, p.248.

23. Remarks on a paper by O.Ore. (Russian). Rev. Math. Pures Appl.1(1956), no.2, 5-12. MR 18, p.275.

24. Rectification a mon travail ”Les ensembles partiellement ordonnes etle theoreme de raffinement de Schreier. II. (Theorie des multistructures)”.Czechoslovak Math. J. 6(81)(1956), 287-288. MR 19, p.243.

25. Sur la theorie generale des produits reguliers. C.R. Acad, Sci. Paris243(1956), 1092-1093. MR 18, p.279.

26. Sur une interpretation topologique de la notion de normalite unitaire.Bull. Sci. Math. 81(1957), Avril-Juin, 87-112. MR 20, 1643.

27. Sur la theorie generale des produits reguliers. C.R. Acad. Sci. Paris244(1957), 1595-1597. MR 19, p.385.

28. Sur la theorie generale des produits reguliers. C.R. Acad. Sci. Paris244(1957), 1702-1704. MR 19, p.528.

29. Sur la theorie generale des produits reguliers. C.R. Acad. Sci. Paris245(1957), 267-270. MR 19, p.528.

30. Uber die allgemeine Theorie der regularen Produkte von Herrn O.N.Golovin. II. Math. Nachr. 16(1957), 137-194. MR 20, 1642.

31. Uber die allgemeine Theorie der regularen Produkte von Herrn O.N.Golovin. III. Math. Nachr. 21(1960), 1-36. MR 22A, 12135.

32. Sur la theorie generale des produits reguliers de Monsieur O.N. Golovine.V.10 Publ. Sci. Univ. Alger Ser. A 4(1957), no.2, 111-143. MR 20, 6470.

33. Sur la fonction de Mobius. C.R. Acad. Sci. Paris 246(1958), 863-865.MR 20, 6370.

34. Sur la theorie generale des ensembles partiellement ordonnes. C.R.Acad. Sci. Paris 247(1958), 2265-2268. MR 20, 5745.

35. Sur la fonction de Mobius. C.R. Acad. Sci. Paris 246(1958), 2553-2555.MR 20, 6371.

36. Bemerkungen zur Theorie der Vielverbande. Math. Nachr. 20(1959),1-16. MR 22A, 19.

10It seems that No.IV of this series has never been published.

32

37. Remarques sur un theoreme de Monsieur Oleg N. Golovine. Czechoslo-vak Math. J. 9(84)(1959), 475-484. MR 23A, 193.

38. Sur la theorie generale des ensembles partiellement ordonnes. Publ. Sci.Univ. Alger Ser. A 7(1960), 5-39. MR 26, 2370.

39. La theorie des multitreillis et son role en algebre et en geometrie. Publ.Sci. Univ. Alger Ser. A 7(1960), 41-58. MR 27, 2447.

40. Bemerkungen zur Theorie der Vielverbande. IV. Uber die Mobius’scheFunktion. Proc. Cambridge Philos. Soc. 56(1960), 291-317. MR 22A, 10929.

41. Uber den Kommutatrixbegriff. Proc. London Math. J. (3) 10(1960),514-530. MR 22A, 11034.

42. Sur la theorie generale des ensembles partiellement ordonnes. I. Proc.Japan Acad. 36(1960), 590-594. MR 24A, 3089.

43. Sur la theorie generale des ensembles partiellement ordonnes. II. Proc.Japan Acad. 36(1960), 595-597. MR 24A, 3089.

44. Sur la theorie generale des ensembles partiellement ordonnes. III. Proc.Japan Acad. 36(1960), 636-638. MR 24A, 3089.

45. Sur une caracterisation abstraite des algebres de Boole. I. C.R. Acad.Sci. Paris 251(1960), 622-623. MR 22A, 7962.

46. Sur une caracterisation abstraite des algebres de Boole. II. C.R. Acad.Sci. Paris 251(1960), 835-836. MR 22A, 7963.

47. Zur abstrakten Begrundung der Fuhrertheorie. Math. Japon. 6(1960/61),1-25. MR 28, 5014.

48. Sur une propriete d’interpolation remarquable dans la theorie des en-sembles partiellement ordonnes. Acta Sci. Math. (Szeged) 22(1961), 1-5. MR24A, 3091.

49. Sur la theorie generale des ensembles partiellement ordonnes. IV. Proc.Japan Acad. 37(1961), 83-84. MR 24A, 3090.

50. On the general theory of partially ordered sets. (Russian). Acta Fac.Rer. Nat. Univ. Comenian. 5(1961), 397-429. MR 27, 61.

51. The theory of multilattices and its significance in algebra and geometry.(Russian). Acta Fac. Rer. Nat. Univ. Comenian. 5(1961), 431-448. MR 24A,1228.

52. Sur la theorie generale des ensembles partiellement ordonnes. I-III.Publ. Sci. Univ. Alger Ser. A 8(1961), 90 pp. MR 27, 5701.

53. Remarques sur la theorie des multitreillis. VI. Mat.-Fyz. Casopis Sloven.Akad. Vied 14(1964), 163-207. MR 31, 94.

54. Geometrische Kontinua und diametrische Raume. Spisy Prırod. Fak.Univ. Purkyne (Brno) Ser. A 29(1965), 469-473. MR 33, 4744.

55. Proprietes d’interpolation des espaces diametriques. Math. Japon.13(1968), 5-19. MR 38, 3774.

56. Remarques sur les subisometries d’un espace diametrique. Mem. Fac.Ci. Habana Ser. Mat. 1(1968), no.7, fasc.6, 11-18. MR 47, 5840.

57. A propos du second theoreme d’isomorphisme pour les espaces diametri-ques. Math. Japon. 13(1968), 21-31. MR 41, 6730.

58. Ensembles ordonnes, fonctions reelles, espaces diametriques. Publ. Inst.Math. (Beograd) (NS) 9(23) (1969), 143-152. MR 40, 63.

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59. Ensembles ordonnes et fonctions monotones. Univ. Lisboa Revista Fac.Ci. A(2) 14(1972/73), 165-194. MR 48, 8313.

Sources of Mihail Benado

R. Baer1945. Representations of groups as quotient groups. I. Trans. Amer. Math.

Soc. 58, 295-346.D. Barbilian

1946. Metrisch-konkave Verbande. Disquisitiones Math. Phys. 5, 3-63.1951. Curs de teoria grupurilor si structurilor. (lito) Univ. Bucuresti.1953. Normalitati locale si integral-involutive. Stud. Cerc. Mat. 4, 29-62.1956. Teoria Aritmetica a Idealelor (ın Inele Necomutative). Ed. Academiei

Romane.1957. Argumentul lui Euclid pentru infinitatea numerelor prime. Stud. Cerc.

Mat. 7, 7-72.G. Birkhoff

1948. Lattice Theory. Amer. Math. Soc.M.L. Dubreil-Jacotin, R. Croisot, L. Lesieur

1953. Lecons sur la Theorie des Treillis,des Structures Algebriques Or-donnees et des Treillis Geometriques. Gauthier-Villars, Paris.H. Fitting

1936. Uber die Existenz gemeinsamer Verfeinerungen bei direkten Produkt-zerlegungen einer Gruppe. Math. Z. 39, 385-400.

1938. Beitrage zur Theorie der Gruppen endlicher Ordnung. Jahresber.Deutscher Math. Verein 48, 77-141.V. Glivenko

1936. Geometrie des systemes de choses normees. Amer. J. Math. 58,799-828; 59(1937), 941.O.N. Golovin

1950. Nilpotent products of groups (Russian). Mat. Sb. (NS) 27, 427-454.1951a. Meta-abelian products of groups (Russian). Mat. Sb. (NS) 28, 431-

444.1951b. On isomorphism problems of nilpotent decompositions of a group.

(Russian). Mat. Sb. (NS) 28, 445-452.O.N. Golovin, N.P. Goldina

1951. Subgroups of free meta-abelian groups (Russian). Mat. Sb. 37(79),323-336.V. Korınek

1941. Das Schreiersche Satz und das Zassenhaussche Verfahren in Verbanden.Vestnik Cesk. Spol. Nauk Trida Mat.-Prırod. Rocnik 1, 29.W. Krull

34

1924. Axiomatische Begrundung der allgemeinen Idealtheorie. Sitz. Phys.-Math. Soz. Erlangen 56, 47-63.A.G. Kurosh

1940. The Jordan-Holder theorem in arbitrary structures. (Russian). Col-lection of Papers Deducated to Acad. D.A. Grave. (Russian). 110-116.

1953. Theory of Groups. Second ed. (Russian). Moscow.O. Ore

1935. On the foundations of abstract algebra. I. Ann. Math. 26, 406-437.1937. On the theorem of Jordan-Holder. Trans. Amer. Math. Soc. 41,

266-275.A.I. Uzkov

1938. On the theorem of Jordan-Holder. Recueil Math. (Moscou) NS 14,31-43.M. Ward, R.P. Dilworth

1939. Residuated lattices. Trans. Amer. Math. Soc. 45, 335-354.H. Zassenhaus

1937. Lehrbuch der Gruppentheorie. Leipzig and Berlin.

Further researchWe are planning a possible sequel of this paper, including references to sev-

eral articles that cite Benado, including more remarks like those in Section C3,and with substantial references to articles that develop the theory of multilat-tices and/or use it as a tool in computer science. We would also refer to somepapers that do not cite Benado, but nevertheless contain ideas in the same direc-tion. Such a study is appropriate in the present context since it will show thathis work, far from remaining isolated, provides remarkable ideas, some of themnot yet explored, for (i) the theory of partially ordered algebraic systems moregeneral than the lattice ordereded algebraic systems, (ii) a more comprehensivetheory of posets suitable for applications in the algebraic theoretical conputerscience, and (iii) a theory of general order structures. Obviously, only a fewexamples can be shortly indicated here, the substantial part remaining for theprojected future paper.

As for (i), the starting point consists of papers [14], [15], already quoted here.We are compiling a bibliography which so far contains about 15 entries referringto multilattices, and among them 9 mention multilattices in their titles, e.g.[Kol3], [Kol4], and [McA1], [McA2], [McA3]. The recent paper [GaP] considersmultilattice groups, quoting [Vai] and [Jak3]. The latter solves a problem raisedin [Vai]; namely, Jakubık shows that every `-group can be embedded into aJ-group (as introduced in [Vai]) which is not a multilattice group.

For (ii) consider again an example, the Plotkin order; cf. [Plot]. Given aposet A and a finite subset U of A, a set V if minimal upper bounds for Uis said to be complete, if for every upper bound x of U , there is y ∈ V withy ≤ x. A characteristic feature for a Plotkin order is that every finite subset has

35

a complete set of minimal upper bounds, i.e., the characteristic requirement inorder to hve an upper semimultilattice. The paper [GuS] studies the theory ofPlotkin orders for semantic domains.

In the last decade, an important school of computer scientists at the Uni-versity of Malaga is very active in focussing a great deal of research on mul-tilattices as introduced by Benado in [12] and [14]-[17]. Their works recovermultilattices for use in very diversified contexts, both theoretical and applied[Mart], [Med]. Since the operations of multi-suprema and multi-infima are nolonger single-valued, their research leads to the theory of hyperstructures as in[Cord2], [Cord3].

As for (iii), the theory of multilattices developed to some extent accordingto [Bir67] is one of the achievements of the school of Kolibiar [Kol1]-[Kol4]and Jakubık [Jak1], [Jak2], [Jak4]. Their joint paper [JK] on isometries ofmultilattice groups has been very influential; see e.g. [Jas]. One should alsonote the connections with universal algebra [Li], [HaLi] and the treatment ofvaluations and distances in directed multilattices [Li2].

AcknowledgementsThis paper is our tribute to Mihail Benado. We were his students, his courses

have determined our interest in the subject and have remained a reference tous.

We thank Dragic Bankovic, Gheorghe Costovici, Ioana and Laurentiu Leustean,who provided several papers of Benado not available in our libraries. Thanksare also due to Traian Serbanuta and Andrei Popescu for references to Math.Sci. Net.

References

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[Bir67] G. Birkhoff, Lattice Theory. Amer. Math. Soc., Providence, RI 1967.

[Cab] I.P. Cabrera, P. Cordero, G. Gutierez, J. Martinez, M. Ojeda-Aciego,Fuzzy congruence relations on nd-groupoids. Intern. J. Comput. Math.86(2009), 1684-1695.

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