Upper triangular matrices over semirings · 2020-06-26 · Upper triangular matrices over semirings...

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Upper triangular matrices oversemiringsA study of their factorization

Nicholas R. Baeth

Franklin & Marshall College

25 06 2020

Motivation

Motivating example

Example (Factorization in T2(Z)•)

We can factor A =

(6 20 5

)∈ T2(Z)• as:

(1 00 5

)(1 10 1

)2(2 00 1

)(3 00 1

)Considering determinants, this and all other factorizations of Ainvolve exactly 3 irreducibles.

Theorem (Bachman-B-Gossell [BBG14])ϕ :

(a b0 c

)7→ (a, c) is a weak transfer homomorphism from

T2(Z)• to (Z\0)n.

Factorization in T2(Z)• is governed entirely by (Z, ·).

Motivating example

We can factor A =

(6 20 5

)∈ T2(Z)• as:

(1 00 5

)(1 10 1

)2(2 00 1

)(3 00 1

Considering determinants, this and all other factorizations of Ainvolve exactly 3 irreducibles.

(a b0 c

T2(Z)• to (Z\0)n.

Motivating example

We can factor A =

(6 20 5

)∈ T2(Z)• as:

(1 00 5

)(1 10 1

)2(2 00 1

)(3 00 1

(a b0 c

T2(Z)• to (Z\0)n.

Motivating example

We can factor A =

(6 20 5

)∈ T2(Z)• as:

(1 00 5

)(1 10 1

)2(2 00 1

)(3 00 1

(a b0 c

T2(Z)• to (Z\0)n.

Motivating example

We can factor A =

(6 20 5

)∈ T2(Z)• as:

(1 00 5

)(1 10 1

)2(2 00 1

)(3 00 1

(a b0 c

T2(Z)• to (Z\0)n.

Factorization in T2(Z)• is governed entirely by (Z, ·).1 25

From Z to N0

Example (Factorization in T2(N0)•)

We can factor A =

(6 20 5

)∈ T2(N0)• as:

(1 00 5

)(1 10 1

)2(2 00 1

)(3 00 1

)The identity is the only unit of T2(N0)•, and this is a product ofexactly 5 irreducible elements.

Moreover, there is no weak transfer homomorphism from T2(N0)•

to any commutative monoid.

Goals:Generalize N0 in Z to semirings of atomic domains.Study factorization in Tn(S)• where S is such a semiring

From Z to N0

We can factor A =

(6 20 5

)∈ T2(N0)• as:

(1 00 5

)(1 10 1

)2(2 00 1

)(3 00 1

The identity is the only unit of T2(N0)•, and this is a product ofexactly 5 irreducible elements.

From Z to N0

We can factor A =

(6 20 5

)∈ T2(N0)• as:

(1 00 5

)(1 10 1

)2(2 00 1

)(3 00 1

From Z to N0

We can factor A =

(6 20 5

)∈ T2(N0)• as:

(1 00 5

)(1 10 1

)2(2 00 1

)(3 00 1

From Z to N0

We can factor A =

(6 20 5

)∈ T2(N0)• as:

(1 00 5

)(1 10 1

)2(2 00 1

)(3 00 1

Definitions and Terminology

Monoids

DefinitionA monoid semigroup with identity; that is, a set S closed withrespect to some (not necessarily commutative) associative,cancellative (a ∗ b = a ∗ c⇒ b = c) operation ∗.An element u ∈ S, a monoid with identity 1, is a unit if thereis v ∈ S with u ∗ v = 1 = v ∗ u.A monoid is reduced if it has only one invertible element.A nonunit a ∈ S is an atom if a = x ∗ y in S implies x or y is aunit.A factorization of x ∈ S is a representation x = a1 ∗ · · · ∗ anwith each ai an atom of S.

DefinitionLet (S, ∗) be a monoid.

S is atomic if every nonunit can be written as a product ofatoms.An ideal in S is a nonempty subset I with S ∗ I, I ∗ S ⊆ I. Theideal I is principal if I = Sa = aS for a ∈ S.The monoid S satisfies the ACCP provided every ascendingchain of principal ideals stabilizes.If x = a1 · · ·a` with ai atoms of S, ` is a length of x. The set ofall possible lengths is the set of lengths L(x).S is a bounded factorization monoid (BFM) if |L(x)| <∞ forall x ∈ S. S is a half-factorial monoid (HFM) if |L(x)| = 1 for allx ∈ S.S is a finite factorization monoid (FFM) if every x ∈ S hasfinitely many factorizations.

Theorem (Anderson-Anderson-Zafrullah [AAZ90])For any commutative monoid (or integral domain), the followingimplications always hold. In general, the arrows are notreversible.

UFM BFM ACCP atomic

Generalizing N0 in Z

A triple (S,+, ·) is a semiring if:(S,+) is a commutative monoid with identity element 0.(S, ·) is a semigroup with identity element 1 6= 0.

Multiplication distributes over addition.

0 · x = 0 for all x ∈ S.

DefinitionA semiring S is an information semialgebra if (S,+) is a reducedmonoid and (S \ 0, ·) is a commutative monoid. S is reduced if(S, ·) is reduced.

Throughout, (S,+, ·) will denote a reduced informationsemialgebra where both (S,+) and (S, ·) are reducedcommutative cancellative monoids.

0 · x = 0 for all x ∈ S.

Information Semialgebras

Example (Information semialgebras)

1. (N0,+, ·)2. N0[

√d] where d is a positive integer.

3. N0[x], a semiring of polynomials with nonnegative integercoecients

4. Any numerical monoid S; a complement-finite additivesubmonoid of N0.

5. Any Puiseux information semialgebra S, where (S,+) isisomorphic to a Puiseux monoid (an additive submonoid ofthe nonnegative cone of rational numbers ) containing 1.

RemarkFactorizations in N0[

√d], N0[x], and Puiseux monoids studied in

[CCMS09] [CF19], and [CGG20], respectively.7 25

Semialgebra atoms

DefinitionLet (S,+, ·) be a reduced information semialgebra.

We denote by A+(S), the additive atoms of S; a ∈ S\0 suchthat a = b + c ⇒ b = 0 or c = 0.A×(S) denotes the multiplicative atoms; a ∈ S\0, 1 suchthat a = bc ⇒ b = 1 or c = 1.

Lemma (B-Gotti [BS20])For an information semialgebra S, the following are equivalent.

1. (S,+) is atomic and |A+(S)| = 1.2. As monoids, (S,+) ∼= (N0,+).3. As semirings, (S,+, ·) ∼= (N0,+, ·).

Semialgebra atoms

DefinitionLet (S,+, ·) be a reduced information semialgebra.

We denote by A+(S), the additive atoms of S; a ∈ S\0 suchthat a = b + c ⇒ b = 0 or c = 0.A×(S) denotes the multiplicative atoms; a ∈ S\0, 1 suchthat a = bc ⇒ b = 1 or c = 1.

Lemma (B-Gotti [BS20])For an information semialgebra S, the following are equivalent.

1. (S,+) is atomic and |A+(S)| = 1.2. As monoids, (S,+) ∼= (N0,+).3. As semirings, (S,+, ·) ∼= (N0,+, ·).

Triangular matrices over infor-mation semialgebras

Basic Results

DefinitionLet S denote a reduced information semialgebra. Then Tn(S)•

denotes the monoid of n× n upper-triangular matrices withnonzero determinant.

Proposition (B-Sampson [BS20])1. If S is a reduced information semialgebra, then Tn(S)• is

reduced, with In the only invertible element.2. If (S,+) and (S, ·) are atomic, then Tn(S)• is atomic and the

atoms of Tn(S)• are the following:I Iij(α) = I + αEij with 1 ≤ i < j ≤ n and α ∈ A+(S).I Iii(a) = I + (a− 1)Eii with 1 ≤ i ≤ n and a ∈ A×(S).

Basic Results

DefinitionLet S denote a reduced information semialgebra. Then Tn(S)•

denotes the monoid of n× n upper-triangular matrices withnonzero determinant.

Proposition (B-Sampson [BS20])1. If S is a reduced information semialgebra, then Tn(S)• is

reduced, with In the only invertible element.2. If (S,+) and (S, ·) are atomic, then Tn(S)• is atomic and the

atoms of Tn(S)• are the following:I Iij(α) = I + αEij with 1 ≤ i < j ≤ n and α ∈ A+(S).I Iii(a) = I + (a− 1)Eii with 1 ≤ i ≤ n and a ∈ A×(S).

PropositionIf S is a reduced information semialgebra, T2(S) is reduced.

Proof.

Suppose(

a1 b10 c1

)(a2 b20 c2

Then a1a2 = 1 = c1c2. Since (S, ·) is reduced, a1 = a2 = c1 = c2 = 1.

Also, 0 = a1b2 + b1c2 = b2 + b1. Since (S,+) is reduced,b1 = b2 = 0 S.

PropositionIf S is a reduced information semialgebra, T2(S) is reduced.

Proof.

Suppose(

a1 b10 c1

)(a2 b20 c2

Then a1a2 = 1 = c1c2. Since (S, ·) is reduced, a1 = a2 = c1 = c2 = 1.

Also, 0 = a1b2 + b1c2 = b2 + b1. Since (S,+) is reduced,b1 = b2 = 0 S.

PropositionIf S is a reduced information semialgebra, the atoms of T2(S) are:

( a 00 1 ) and ( 1 0

0 a ) with a ∈ A×(S)

( 1 α0 1 ) with α ∈ A+(S)

Proof.For any a,b, c ∈ S, note that

(a b0 c

(1 00 c

)(1 b0 1

)(a 00 1

)With c = c1 · · · cm, b = b1 + · · ·+ bl, and a = a1 · · ·ak with eachai, cj ∈ A×(S) and each bi ∈ A+(S), we can further factor A as:(

1 00 c1

)· · ·(

1 00 cm

)(1 b10 1

)· · ·(

1 bl0 1

)(a1 00 1

)· · ·(

ak 00 1

PropositionIf S is a reduced information semialgebra, the atoms of T2(S) are:

( a 00 1 ) and ( 1 0

0 a ) with a ∈ A×(S)

( 1 α0 1 ) with α ∈ A+(S)

Proof.For any a,b, c ∈ S, note that

(a b0 c

(1 00 c

)(1 b0 1

)(a 00 1

)With c = c1 · · · cm, b = b1 + · · ·+ bl, and a = a1 · · ·ak with eachai, cj ∈ A×(S) and each bi ∈ A+(S), we can further factor A as:(

1 00 c1

)· · ·(

1 00 cm

)(1 b10 1

)· · ·(

1 bl0 1

)(a1 00 1

)· · ·(

ak 00 1

)11 25

DefinitionA monoid is transfer-Krull provided there is a (weak) transferhomomorphism to a commutative Krull monoid.

Proposition (B-Sampson [BS20])For n ≥ 2 and S a reduced information semialgebra, Tn(S)• is nottransfer-Krull

Proof.Suppose ϕ : T2(S)• → M is a (weak) transfer homomorphism to acommutative cancellative monoid. Then, with α ∈ A+(S),

( 2 00 1 )( 1 α

0 1 ) = ( 1 α0 1 )2( 2 0

0 1 )⇒ ϕ( 2 00 1 )ϕ( 1 α

0 1 ) = ϕ( 2 00 1 )ϕ( 1 α

0 1 )2.

Thus so ϕ( 1 α0 1 ) = 1M, a contradiction.

DefinitionA monoid is transfer-Krull provided there is a (weak) transferhomomorphism to a commutative Krull monoid.

Proposition (B-Sampson [BS20])For n ≥ 2 and S a reduced information semialgebra, Tn(S)• is nottransfer-Krull

Proof.Suppose ϕ : T2(S)• → M is a (weak) transfer homomorphism to acommutative cancellative monoid. Then, with α ∈ A+(S),

( 2 00 1 )( 1 α

0 1 ) = ( 1 α0 1 )2( 2 0

0 1 )⇒ ϕ( 2 00 1 )ϕ( 1 α

0 1 ) = ϕ( 2 00 1 )ϕ( 1 α

0 1 )2.

Thus so ϕ( 1 α0 1 ) = 1M, a contradiction.

Factorization in Tn(S)•

Divisor-closed submonoids

DefinitionA submonoid S′ of a monoid S is divisor-closed provided thatwhenever x ∈ S and y ∈ S′ and x divides y, x ∈ S′.

Theorem (B-Gotti [BG])Let S be a reduced information semialgebra and let n ≥ 2. Thefollowing are divisor-closed submonoids of Tn(S)•.

Un(S) = A = (aij) ∈ Tn(S)• : aii = 1∀ iI + (s− 1)Eii : s ∈ S• ∼= (S•, ·)I + sEij : s ∈ S ∼= (S,+)

I + sEij : s ∈ S is also divisor-closed in Un(S).

Reduction to the 2× 2 case

Proposition (B-Sampson [BS20])Let S be a reduced information semialgebra, set n > 2, and defineϕ : T2(S)• → Tn(S)• by

(a b0 c

a b0 c 00 In−2

Then T2(S)• ∼= ϕ(T2(S)•), a divisor-closed submonoid of Tn(S)•.

Consequently, many proofs can be reduced immediately to the2× 2 case.

Half-factoriality

Proposition (B-Sampson [BS20])Tn(S)• is not an HFM for any reduced information semialgebra Sand any n ≥ 2.

Proof.(S,+) ∼= I + sE12 : s ∈ S is divisor closed in Tn(S)• so if (S,+) isnot atomic, neither is Tn(S)•.

If (S,+) is atomic, then 1 ∈ A+(S). Otherwise, 1 = x + y for somex, y ∈ S• implies s = s(x + y) = sx + sy ∈ S• + S• for all s ∈ S•.

(1 00 m

)(1 10 1

(1 10 1

)(1 00 m

Now ` ∈ LT2 (( 1 00 m )) implies `+ 1, `+ m ∈ LT2(A).

Half-factoriality

(1 00 m

)(1 10 1

(1 10 1

)(1 00 m

Half-factoriality

(1 00 m

)(1 10 1

(1 10 1

)(1 00 m

Bounded Factorizations

ExampleClearly (N0,+) and (N, ·) are BFMs. For A ∈ Tn(N0)•,max L(A) = Ω(det(A)) + Σ(A) <∞ and so Tn(N0)• is a BFM.

ExampleLet S be the additive submonoid of Q≥0 generated by the setA = (2/3)n : n ∈ N0. S is a redued information semialgebra.(S,+) is an atomic monoid with set of atoms A.For every n ∈ N, the 3 = 2

( 23)n

i=0( 2

3)i and so (S,+) is not a

BFM. Moreover, T3(S)• is not a BFM since( 1 0 30 1 00 0 1

0 1 00 0 1

)2( 1 0 10 1 00 0 1

)( 1 0 23

0 1 00 0 1

)· · ·(

0 1 00 0 1

)for every n ∈ N.

ExampleLet S be the additive submonoid of Q≥0 generated by the setA = (2/3)n : n ∈ N0. S is a redued information semialgebra.

(S,+) is an atomic monoid with set of atoms A.For every n ∈ N, the 3 = 2

( 23)n

i=0( 2

0 1 00 0 1

)2( 1 0 10 1 00 0 1

)( 1 0 23

0 1 00 0 1

)· · ·(

0 1 00 0 1

)for every n ∈ N.

ExampleLet S be the additive submonoid of Q≥0 generated by the setA = (2/3)n : n ∈ N0. S is a redued information semialgebra.(S,+) is an atomic monoid with set of atoms A.

For every n ∈ N, the 3 = 2( 2

i=0( 2

0 1 00 0 1

)2( 1 0 10 1 00 0 1

)( 1 0 23

0 1 00 0 1

)· · ·(

0 1 00 0 1

)for every n ∈ N.

ExampleLet S be the additive submonoid of Q≥0 generated by the setA = (2/3)n : n ∈ N0. S is a redued information semialgebra.(S,+) is an atomic monoid with set of atoms A.For every n ∈ N, the 3 = 2

( 23)n

i=0( 2

0 1 00 0 1

)2( 1 0 10 1 00 0 1

)( 1 0 23

0 1 00 0 1

)· · ·(

0 1 00 0 1

)for every n ∈ N.

Finite Factorizations

ExampleClearly (N0,+) and (N, ·) are FFMs. For A ∈ Tn(N0)•, there are atmost Ω(det(A)) + (n2 − n)/2 atoms that divide it, each appearingat most finitely many times, and so Tn(N0)• is a FFM.

ExampleThe subsemiring S = 0, 1 ∪Q≥2 of Q is a reduced informationsemialgebra with A×(S) = [2, 4) ∩Q.

In particular, for all n ∈ N, an = 3nn+1 and bn = 3(n+1)

n aremultiplicative atoms of S.

T2(S)• is not an FFM since ( 9 00 1 ) = ( an 0

0 1 )(

bn 00 1)

for all n ∈ N.

0 1 )(

bn 00 1)

for all n ∈ N.

0 1 )(

bn 00 1)

for all n ∈ N.

0 1 )(

bn 00 1)

for all n ∈ N.

Two operations

RemarkIf D is an atomic domain, the arithmetic of Tn(D)• is completelydetermiend by that of (D•, ·).

DefinitionIf S is a reduced information semialgebra and ∈ FFM, BFM, ACCP, atomic, then we say that S is a bi-provided that both its additive monoid (S,+) and multiplicativemonoid (S•, ·) are .

As we have observed, the arithmetic of Tn(S)• depends on thearithmetic of both (S,+) and (S, ·).

Factorization in Tn(S)•

Theorem (B-Gotti [BG])Let S be a reduced information semialgebra. For n ≥ 2, eachimplication in the following diagram holds.

Tn(S)• is FFM Tn(S)• is BFM Tn(S)• satisfies ACCP Tn(S)• is atomic

S is bi-FFM S is bi-BFM S satisfies bi-ACCP S is bi-atomic

(S,+) is FFM (S,+) is BFM (S,+) satisfies ACCP (S,+) is atomic

Un(S) is FFM Un(S) is BFM Un(S) satisfies ACCP Un(S) is atomic

Moreover, none of the horizontal implications is, in general,reversible. Finally, Tn(S)• is never an HFM when n ≥ 2.

Arithmetical results:a focus on Tn(N0)

DefinitionAn atom is almost prime-like if whenever it appears in onefactorization of an element it appears in all factorizations ofthat element.

An almost prime-like element is prime-like if for each fixedelement A, it always appears with the same multiplicity inany fatorization of A.

An atom A is absolutely irreducible if no other atoms divideAn for any n ∈ N.

Theorem(B-Sampson-Chen-Liu-Heilbrunn-Young [BS20, BC+])Consider the monoid Tn(N0)• for some n ≥ 2.

The atoms are:I Iij(1) = I + Eij with 1 ≤ i < j ≤ n.I Iii(p) = I + (p− 1)Eii with 1 ≤ i ≤ n and p prime in N.

Atoms of the form Iii(p) are prime-like.Atoms of the form Iij(1) are almost prime-like but notprime-like if j = i + 1.Atoms of the form Iij(1) are not almost prime-like if j > i + 1.All atoms of Tn(N0)• are absolutely irreducible.

Delta set

DefinitionIf L(A) = `1 < `2 < · · · , ∆(A) = `i+1 − `i : i ≥ 1.

Proposition (B-Sampson [BS20])∆(Tn(N0)•) = ∪∆(A) = N for all n ≥ 2.

Proof.Fix k ∈ N and choose a prime p > 2k + 2. Set M =

( p 2p−2k−20 2

Atoms dividing M: A = ( 1 00 2 ), B = ( 1 1

0 1 ), C =( p 0

0 1). Factorizations:

AB2p−2k−2C = BAB2p−2k−4 = · · · = Bp−k−1AC and

ABp−2k−2CB = BABp−2k−4CB = · · · = Bp−2k−3

2 ABCB