For Review Only Monoid of Linear Hypersubstitutions for Algebraic Systems of Type ((n),(2)) and its Regularity Journal: Songklanakarin Journal of Science and Technology Manuscript ID SJST-2018-0121.R1 Manuscript Type: Original Article Date Submitted by the Author: 29-Jun-2018 Complete List of Authors: Kumduang, Thodsaporn; Chiang Mai University, Mathematics Leeratanavalee, Sorasak ; Chiang Mai University, Mathematics Keyword: algebraic systems, linear terms, linear formulars, linear hypersubstitutions For Proof Read only Songklanakarin Journal of Science and Technology SJST-2018-0121.R1 Kumduang
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For Review Only
Monoid of Linear Hypersubstitutions for Algebraic Systems
of Type ((n),(2)) and its Regularity
Journal: Songklanakarin Journal of Science and Technology
Manuscript ID SJST-2018-0121.R1
Manuscript Type: Original Article
Date Submitted by the Author: 29-Jun-2018
Complete List of Authors: Kumduang, Thodsaporn; Chiang Mai University, Mathematics Leeratanavalee, Sorasak ; Chiang Mai University, Mathematics
Keyword: algebraic systems, linear terms, linear formulars, linear hypersubstitutions
For Proof Read only
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For Review Only
1
Original Article
Monoid of Linear Hypersubstitutions for Algebraic Systems
of Type ((��, (2�� and its Regularity
Thodsaporn Kumduang1, and Sorasak Leeratanavalee
1,2,*
1 Department of Mathematics, Faculty of Science, Chiang Mai University,
Chiang Mai 50200, Thailand
2 Centre of Excellence in Mathematics, CHE, Bangkok 10400, Thailand
An algebraic system is a structure which consists of a nonempty set together
with a sequence of operations and a sequence of relations on this set. The properties of
this structure are expressed by terms and formulas. In this paper, we show that the set of
all linear hypersubstitutions for algebraic systems of the type ((��, (2�� with a binary
operation on this set and the identity element forms a monoid. Finally, we characterize
idempotent and regular elements on the monoid.
Keywords: algebraic systems, linear terms, linear formulas, linear hypersubstitutions
1. Introduction
The concept of an algebraic system was first introduced by A.I. Mal'cev in 1973.
For approach to algebraic systems, we need some preparations.
Let � be a nonempty set and � ∈ ℕ ≔ ℕ ∖ {0}. An �-ary operation on � is a mapping �: �� → �. We call � the arity of �. An �-ary relation on � is a relation
� ⊆ �� and call � the arity of �. Let �, � be indexed sets and let (����∈� , (����∈� be sequences of operation
symbols and relation symbols, respectively. Let � = (����∈� and �́ = (����∈� where �� has the arity �� for every � ∈ � and �� has the arity �� for every ∈ �.
Definition 1.1 (Mal’cev, 1973) An algebraic system of type (�, �́� is a triple ! ≔(�, (����∈� , (����∈�� consisting of a nonempty set�, a sequence (����∈� of operations on
� where �� is ��-ary for � ∈ � and a sequence (����∈� of relations on � where �� is ��-
ary for ∈ �. The pair (�, �́� is called the type of an algebraic system.
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To classify algebras into collections called varieties we need terms and some
pairs of terms, i.e. equations. To classify algebraic systems into subclasses by logical
sentences we need a language, i.e. quantifier free formulas.
Now, we recall basic notions related to terms. For a natural number � ≥ 1, let %� = {&', … , &�} be a finite set of variables, and let % ≔ ⋃ %��*' = {&', … , &�, … } be countably infinite. Let {�� |� ∈ �} be a set of operation symbols which is disjoint from %. An �-ary term of type � is defined inductively as follows: (i) Every variable &� ∈ %� is an �-ary term of type �. (ii) If,', … , ,�- are �-ary terms of type � and �� is an ��-ary operation symbol,
then ��(,', … , ,�-� is an �-ary term of type �. Let ./(%�� be the set of all �-ary terms of type � which contains &', … , &� and is closed
under finite application of (ii).
Not all of the terms in the second-order language will used to express properties
of algebraic systems. The one is called formulas, first introduced by A.I. Mal'cev in
1973. For approach to formulas see also (Mal’cev, 1973), and we recall the definition of
formula which is defined by Denecke and Phusanga (2008).
To define the quantifier free formulas of type (�, �́� we need the logical connectives ¬ (negation), ∨ (disjunction) and the equation symbol ≈ .
Definition 1.2 (Denecke & Phusanga, 2008) Let � ∈ ℕ. An �-ary quantifier free
formula of type (�, �́� (for simply, formula) is defined in the following steps: (i) If ,', ,0 are �-ary terms of type τ , then the equation 1 2t t≈ is an �-ary
quantifier free formula of type (�, �́�. (ii) If ∈ � and ,', … , ,�1 are �-ary terms of type τ and ��is an ��-ary relation
symbol, then ��(,', … , ,�1� is an �-ary quantifier free formula of type (�, �́�. (iii) If 2 is an �-ary quantifier free formula of type (�, �́�, then ¬2 is an �-ary
quantifier free formula of type (�, �́�. (iv) If 2' and 20 are �-ary quantifier free formulas of type (�, �́�, then 2' ∨ 20 is
an �-ary quantifier free formula of type (�, �́�. Let ℱ(/,/́�(./(%��� be the set of all �-ary quantifier free formulas of type (�, �́�.
In 2012, M. Couceiro and E. Lehtonen introduced the concept of a linear term,
i.e., a term which each variable occurs only once.
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Definition 1.3 (Couceiro & Lehtonen, 2012) An �-ary linear term of type � is defined inductively as follows:
(i) Every &� ∈ %� is an �-ary linear term of type �. (ii) If ,', … , ,�- are �-ary linear terms of type � with 678(,9� ∩ 678(,;� = ∅ for
all 1 ≤ > < @ ≤ �� (where 678(,� is the set of all variables occurring in the the term ,) and �� is an ��-ary operation symbol, then ��(,', … , ,�-� is an �-ary linear term of type �.
Let ./9��(%�� be the set of all �-ary linear terms of type �. In this paper, we consider an algebraic system of type A(��, (2�B, i.e., we have
only one �-ary operation symbol and one binary relation symbol. We define the new definition of linear formulas of type ((��, (2�� and give the concept of superposition of linear terms and superposition of linear formulas. This leads to introduce the concept of
linear hypersubstitutions for algebraic systems of type ((��, (2��. We show that the set of all linear hypersubstitutions for algebraic systems of type A(��, (2�B together with a binary operation ∘D and an identity element forms a monoid. Furthermore, the characterizations of idempotent and regular elements are investigated.
2. Linear Terms of Type (E� and Linear Formulas of Type A(E�, (F�B
Let 678(,� be the set of all variables occurring in the term , and let 678(2� be the set of all variables occurring in the formula 2.
In this section, we first defined the definition of a linear term and a quantifier
free linear formula of type A(��, (2�B as follows: Definition 2.1 Let G, � ∈ ℕ with G ≥ �. An G-ary linear term of type (�� is defined in the following inductive way:
(i) Every &� ∈ %H is an G-ary linear term of type (��. (ii) If ,', … , ,� are G-ary linear terms of type (�� with 678(,9� ∩ 678(,;� = ∅
for all 1 ≤ > < @ ≤ � and � is an �-ary operation symbol, then �(,', … , ,�� is an G-ary linear term of type (��.
Let .(��9��(%H� be the set of all G-ary linear terms of type (��. Example 2.2 Let (�� = (2� be the type with a binary operation symbol � and %0 ={&', &0}. Then &', &0, �(&', &0�, �(&0, &'�are examples of binary linear terms of type (2�.
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Definition 2.3 Let G, � ∈ ℕ with G ≥ �. An G-ary quantifier free linear formula of type A(��, (2�B (for simply, linear formula) is defined in the following steps: (i) If ,', ,0 are G-ary terms of type (�� and 678(,'� ∩ 678(,0� = ∅ then the
equation ,' ≈ ,0 is an G-ary quantifier free formula of type A(��, (2�B. (ii) If ,', ,0 are G-ary terms of type (�� with 678(,'� ∩ 678(,0� = ∅ and � is a
binary relation symbol, then �(,', ,0� is an G-ary quantifier free formula of type A(��, (2�B.
(iii) If 2 is an G-ary quantifier free formula of type A(��, (2�B, then F¬ is an G-ary quantifier free formula of type A(��, (2�B.
(iv) If 2' and 20 are G-ary quantifier free formulas of type A(��, (2�B, then 2' ∨ 20 is an G-ary quantifier free formula of type A(��, (2�B.
Let ℱA(��,(0�B9�� (.(��(%H�� be the set of all G-ary quantifier free linear formulas of type
A(��, (2�B. Remark 2.4 The linear formulas defined by (i) and (ii) are called atomic linear
formulas.
Example 2.5 Let ((2�, (2�� be a type, i.e., we have one binary operation symbol � and one binary relation symbol � and let %0 = {&', &0}. Then the binary atomic linear formulas of type ((2�, (2�� are &' ≈ &0, &0 ≈ &', �(&', &0�, �(&0, &'�. Moreover, we obtained all other linear formulas of type ((2�, (2�� from binary atomic linear formulas of type ((2�, (2�� by using the connectives ¬ and ∨.
Next, we give the concepts of the superposition of linear terms and linear
formulas for algebraic systems of type A(��, (2�B. For convenient, we let J� be the set of all permutations of {1, … , �}.
3. Superposition of Linear terms and Linear Formulas
Definition 3.1 Let G, � ∈ ℕ with G ≥ �, , ∈ .(��9��(%��and ,', … , ,� ∈.(��9��(%H�with 678(,9� ∩ 678(,;� = ∅ for all 1 ≤ > < @ ≤ �. We define the concept of a superposition of linear terms
JH9��� :.(��9��(%�� × (.(��9��(%H��� ⊸→ .(��9��(%H� as follows:
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Now, we can extend the concept of this superposition to quantifier free linear
formulas by substituting variables occurring in a quantifier free linear formula by a
linear term, and obtain a new quantifier free linear formula. We explain this by the
following operations PH9��� when G, � ≥ 1. Definition 3.2 Let G, � ∈ ℕ with G ≥ �and ,', … , ,� ∈ .(��9��(%H�with 678(,9� ∩678(,;� = ∅ for all 1 ≤ > < @ ≤ �. The operation
.(��A%VB, 678(,9� ∩ 678(,;� = ∅ for all 1 ≤ > < @ ≤ T and W', … , W� ∈
.(��(%��, 678(W9� ∩ 678(W;� = ∅ for all 1 ≤ > < @ ≤ �. (FC2) P�9��� (U, &', … , &�� = U.
4. Monoid of Linear Hypersubstitutions for Algebraic Systems of Type A(E�, (F�B
In this section, we would like to form the new structure of so-called "Monoid of
Linear Hypersubstitutions for Algebraic Systems of Type A(��, (2�B". The way to approach this, we first define the based set.
Definition 4.1 Let � ∈ ℕ. A linear hypersubstitution for algebraic systems of type
A(��, (2�B is a mapping X: {�} ∪ {�} → .(��9��(%�� ∪ℱA(��,(0�B
9�� R.(��(%0�S which maps an �-ary operation symbol � to an �-ary linear term of type (��and maps a binary relation symbol � to a binary quantifier free linear formula of type A(��, (2�B. We denote the set of all linear hypersubstitutions for algebraic systems of type
A(��, (2�B by YZT9��A(��, (2�B.
From now on, every element in YZT9��A(��, (2�B will be denoted by X[,\, that
means X[,\(�� = , and X[,\(�� = 2. To define a binary operation on YZT9��A(��, (2�B, we extend a linear
hypersubstitution for algebraic systems X to a mapping X] defined by the following definition.
Definition 4.2 Let X[,\ ∈ YZT9��A(��, (2�B,O ∈ J� and ^ ∈ J0. Then we define a
mapping
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(i) X][,\[&�]:=&� for every � = 1,… , �. (ii) X][,\[�(&N('�, … , &N(���]≔J�9���AX[,\(��, X][,\a&N('�b, … , X][,\a&N(��bB. (iii) X][,\a&c('� ≈ &c(0�b ≔ X][,\a&c('�b ≈ X][,\a&c(0�b. (iv) X][,\[�(&c('�, &c(0��]≔P09��0 (X[,\(��, X][,\a&c('�b, X][,\a&c(0�b�. (v) X][,\[¬2] ≔ ¬X][,\[2] for 2 ∈ ℱA(��,(0�B
9�� R.(��(%0�S. Now, we define a binary operation ∘D on YZT9��A(��, (2�B as follows:
Definition 4.3 Let X[d,\d , X[e,\e ∈ YZT9��A(��, (2�B and ∘ be the usual composition of mapping. Then we define a binary operation ∘D on YZT9��A(��, (2�B by
X[d,\d ∘D X[e,\e ≔ X][d,\d ∘ X[e,\e.
Next, we prove that a binary operation as we already defined in Definition 4.3
satisfies associative law. To get our result, we need some preparations as follows:
Lemma 4.4 For each X[,\ ∈ YZT9��A(��, (2�B, O ∈ J� and ^ ∈ J0. Then we have
Proof. (i) Let , ∈ .(��9��(%��. We give a proof by induction on the complexity of a linear term ,. Obviously, if , = &� for all 1 ≤ � ≤ �. If , = �(&N('�, … , &N(��� and for every > = 1, … , � we assume that X][,\aJ�9���A&N(9�, &N('�, … , &N(��Bb = J�9���(X][,\[&N(9�], X][,\a&N('�b, … , X][,\a&N(��b�, then by Theorm 3.3 we get X][,\aJ�9���A�(&N('�, … , &N(���, &N('�, … , &N(��Bb = X][,\a�(J�9���A&N('�, &N('�, … , &N(��B,… , �(J�9���A&N(��, &N('�, … , &N(��B�b =J�9���(X[,\(��, X][,\aJ�9���A&N('�, &N('�, … , &N(��Bb, … , X][,\[J�9���A&N(��, &N('�, … , &N(��B]� =J�9���(X[,\(��, J�9���(X][,\[&N('�], X][,\a&N('�b, … , X][,\a&N(��b�, …, J�9���(X][,\[&N(��], X][,\a&N('�b, … , X][,\a&N(��b�. =J�9���(J�9���AX[,\(��, X][,\a&N('�b, … , X][,\a&N(��bB, X][,\a&N('�b, … , X][,\[&N(��]� =J�9���(X][,\[�(&N('�, … , &N(���], X][,\a&N('�b, … , X][,\a&N(��b). (ii) Let U ∈ ℱA(��,(0�B
9�� R.(��(%0�S. We give a proof by the following steps.
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If U has the form &c('� ≈ &c(0�, then we have X][,\aP09��0A&c('� ≈ &c(0�, &c('�, &c(0�Bb = X][,\aJ09��0A&c('�, &c('�, &c(0�B ≈ J09��0A&c(0�, &c('�, &c(0�B�b = J09��0(X][,\[&c('�],X][,\[&c('�],X][,\[&c(0�])≈ J09��0(X][,\[&c(0�],X][,\[&c('�],X][,\[&c(0�])
=P09��0(X][,\[&c('�] ≈ X][,\[&c(0�], X][,\[&c('�], X][,\[&c(0�]� =P09��0(X][,\[&c('� ≈ &c(0�], X][,\[&c('�], X][,\[&c(0�]�. If U has the form �(&c('�, &c(0��, then by Theorem 3.3 we have X][,\aP09��0A�(&c('�, &c(0��, &c('�, &c(0�Bb = X][,\a�(J09��0A&c('�, &c('�, &c(0�B, �(J09��0A&c(0�, &c('�, &c(0�B�b =P09��0(X[,\(��, X][,\aJ09��0A&c('�, &c('�, &c(0�Bb, X][,\[J09��0A&c(0�, &c('�, &c(0�B]� =P09��0(X[,\(��, J09��0(X][,\[&c('�], X][,\a&c('�b, X][,\a&c(0�b�, J09��0(X][,\[&c(0�], X][,\a&c('�b, X][,\a&c(0�b�. =P09��0(P09��0AX[,\(��, X][,\a&c('�b, X][,\a&c(0�bB, X][,\a&c('�b, X][,\[&c(0�]� =P09��0 (X][,\[�(&c('�, &c(0��], X][,\a&c('�b, X][,\[&c(0�]). If U has the form ¬2 and assume that X][,\aP09��0A2, &c('�, &c(0�Bb = P09��0(X][,\[2], X][,\a&c('�b, X][,\a&c(0�b�, then we get X][,\aP09��0A¬2, &c('�, &c(0�Bb = X][,\a¬(P09��0A2, &c('�, &c(0�B�b =¬(X][,\aP09��0A2, &c('�, &c(0�Bb� = ¬(P09��0AX][,\[2], X][,\a&c('�b, X][,\a&c(0�bB =P09��0(¬X][,\[2], X][,\a&c('�b, X][,\a&c(0�b� = P09��0(X][,\[¬2], X][,\a&c('�b, X][,\a&c(0�b�.
As a result of Lemma 4.4, we have the following lemma.
Lemma 4.5 Let X[d,\d , X[e,\e ∈ YZT9��A(��, (2�B. Then we have (X[d,\d ∘D X[e,\e�∧ = X][d,\d ∘ X][e,\e .
Proof. Let , ∈ .(��9��(%��, we give a proof by induction on the complexity of a linear term ,. If , = &� ; 1 ≤ � ≤ �, then AX[d,\d ∘D X[e,\eB
∧[&�] = &� = X][d,\d[&�] = X][d,\d[X][e,\e[&�]] = (X][d,\d ∘ X][e,\e�[&�]. If , = �(&N('�, … , &N(���, then by Lemma 4.4 we have that AX[d,\d ∘D X[e,\eB
If U has the form ¬2 and assume that (X[d,\d ∘D X[e,\e�∧[2] = (X][d,\d ∘ X][e,\e�[2], then we obtain that (X[d,\d ∘D X[e,\e�∧[¬2] = ¬(X[d,\d ∘D X[e,\e�∧[2] = ¬AX][d,\d ∘X][e,\eB[2] = ¬X][d,\d gX][e,\e[2]h = X][d,\d g¬X][e,\e[2]h = X][d,\d gX][e,\e[¬2]h = (X][d,\d ∘X][e,\e�[¬2].
It follows from Lemma 4.5 that the binary operation ∘D satisfies associative law. We prove this fact in the next lemma.
Lemma 4.6 For any X[d,\d , X[e,\e , X[i,\i ∈ YZT9��A(��, (2�B, we have AX[d,\d ∘D X[e,\eB ∘D X[i,\i = X[d,\d ∘D (X[e,\e ∘D X[i,\i�.
Proof. By using Lemma 4.5 and the fact that ∘ satisfies associative law, it can be shown that ∘D satisfies associative law. In fact, we have AX[d,\d ∘D X[e,\eB ∘D X[i,\i =
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(X[e,\e ∘D X[i,\i� = X[d,\d ∘D (X[e,\e ∘D X[i,\i�. Let X�j be a linear hypersubstitution for algebraic systems which maps the
operation symbol � to the linear term �(&', … , &�� and maps the relation symbol � to the linear formula �(&', &0�, i.e. X�j(�� = �(&', … , &�� and X�j(�� = �(&', &0�.
A linear hypersubstitution X�j is claimed to be an identity, which we will prove this fact in the next lemma.
Lemma 4.7 For any linear term , ∈ .(��9��(%��and linear formula U ∈ ℱA(��,(0�B
9�� R.(��(%0�S, we have X]�j[,] = , and X]�j[U] = U. Proof. Let , ∈ .(��9��(%��, we give a proof by induction on the complexity of a linear term ,. If , = &� with � = 1,… , �, then X]�j[&�] = &�. If , = �A&N('�, … , &N(��B where O ∈ J�, then we get X]�ja�A&N('�, … , &N(��Bb = J�9���(X[,\(��, X][,\a&N('�b, … , X][,\a&N(��b) = J�9���A�(&', … , &��, &N('�, … , &N(��B = �A&N('�, … , &N(��B. Next, let U ∈ℱA(��,(0�B
9�� R.(��(%0�S , ^ ∈ J0, we give a proof by the following steps.
If U has the form &c('� ≈ &c(0�, then we have X]�ja&c('� ≈ &c(0�b = X]�ja&c('�b ≈X]�ja&c('�b = &c('� ≈ &c(0�. If U has the form �(&c('�, &c(0��, then X]�ja�(&c('�, &c(0��b = P09��0AX[,\(��, X][,\a&c('�b, X][,\a&c(0�bB =P09��0A�(&', &0�, &c('�, &c(0�B = �(&c('�, &c(0��. If U has the form ¬2 and assume that X]�j[2] = 2, then X]�j[¬2] = ¬X]�j[2] = ¬2. Lemma 4.8 Let X�j ∈ YZT9��A(��, (2�B. Then we have X�j is an identity element
with respect to ∘D.
Proof. First, we prove that X�j is a left identity element by using Lemma 4.7. Let
X[,\ ∈ YZT9��A(��, (2�B. Then we have AX�j ∘D X[,\B(�� = AX]�j ∘ X[,\B(�� =X]�jaX[,\(��b = X[,\(�� and AX�j ∘D X[,\B(�� = AX]�j ∘ X[,\B(�� = X]�jaX[,\(��b = X[,\(��. Now, we show that X�j is a right identity element. Let X[,\ ∈ YZT9��A(��, (2�B. By Theorem 3.3, we obtain that AX[,\ ∘D X�jB(�� = AX][,\ ∘ X�jB(�� = X][,\[X�j(��] =X][,\[�(&', … , &��] = J�9���AX[,\(��, X][,\[&'], … , X][,\[&']B =J�9���AX[,\(��, &', … , &�B = X[,\(�� and AX[,\ ∘D X�jB(�� = AX][,\ ∘ X�jB(�� =X][,\[X�j(��] = X][,\[�(&', &0�] = P09��0AX[,\(��, X][,\[&'], X][,\[&0]B =
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P09��0AX[,\(��, &', &0B = X[,\(��. This implies that X�j ∘D X[,\ = X[,\ = X[,\ ∘D X�j.
Therefore, X�j is an identity element.
Theorem 4.9 ℋZT9��A(��, (2�B ≔ (YZT9��A(��, (2�B,∘D, X�j� is a monoid. Proof. From Lemma 4.6 and 4.8, the conclusion holds.
5. Idempotent and Regular Elements in lmnopEA(E�, (F�B
In this section we study some semigroup properties of ℋZT9��A(��, (2�B, especially we characterize idempotency and regularity of X[,\ ∈ YZT9��A(��, (2�B. We first introduce some notations and definitions of idempotent and regular elements in
YZT9��A(��, (2�B with respect to ∘D.
For any X[,\ ∈ YZT9��A(��, (2�B, O ∈ J�, ^ ∈ J0 we denote :
q' ≔ rX[,\ |, = &� ∈ %�, 2 = &c('� ≈ &c(0�s, q0 ≔ rX[,\ |, = &� ∈ %�, 2 = �(&c('�, &c(0��s, qt ≔ rX[,\ |, = &� ∈ %�, 2 = ¬A&c('� ≈ &c(0�Bs, qu ≔ rX[,\ |, = &� ∈ %�, 2 = ¬�(&c('�, &c(0��s, qv ≔ rX[,\ |, = �A&N('�, … , &N(��B, 2 = &c('� ≈ &c(0�s, qw ≔ rX[,\ |, = �A&N('�, … , &N(��B, 2 = �(&c('�, &c(0��s, qx ≔ rX[,\ |, = �A&N('�, … , &N(��B, 2 = ¬A&c('� ≈ &c(0�Bs, qy ≔ rX[,\ |, = �A&N('�, … , &N(��B, 2 = ¬�(&c('�, &c(0��s. We note that z = {q', … , qy} is a partition of YZT9��A(��, (2�B. The concepts of an idempotent element and a regular element are defined in
ℋZT9��A(��, (2�B. An element X[,\ ∈ YZT9��A(��, (2�B is said to be idempotent if
X[,\ ∘D X[,\ = X[,\, that is, (X[,\ ∘D X[,\�(�� = X[,\(�� and (X[,\ ∘D X[,\�(�� = X[,\(��. And X[,\ ∈ YZT9��A(��, (2�B is called regular if there is an element X[{ ,\{ ∈YZT9��A(��, (2�B such that X[,\ = X[,\ ∘D X[{ ,\{ ∘D X[,\. The semigroup ℋZT9��A(��, (2�B is called regular if every element in YZT9��A(��, (2�Bis regular. Furthermore, we denote the set of all idempotent and regular in ℋZT9��A(��, (2�B by | RYZT9��A(��, (2�BS and P}~ RYZT9��A(��, (2�BS, respectively.
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Lemma 5.1 (Burris, 1981) Suppose 2 is a formula in some ℱ(/,/́�(./(%���. Then the following pair of formula is equivalent: ¬(¬2� ≡ 2. Lemma 5.2 Let X[,\ ∈ YZT9��A(��, (2�B. Then X[,\ is idempotent if and only if
X][,\[,] = , and X][,\[2] = 2. Proof. Assume that X[,\ is idempotent, i.e., (X[,\ ∘D X[,\�(�� = X[,\(�� and (X[,\ ∘D X[,\�(�� = X[,\(��. We now consider X][,\[,] = X][,\aX[,\(��b = AX][,\ ∘X[,\B(�� = AX[,\ ∘D X[,\B(�� = X[,\(�� = , and X][,\[2] = X][,\aX[,\(��b =AX][,\ ∘ X[,\B(�� = AX[,\ ∘D X[,\B(�� = X[,\(�� = 2. Conversely, let X][,\[,] = , and X][,\[2] = 2. Then we haveAX[,\ ∘D X[,\B(�� = AX][,\ ∘ X[,\B(�� = X][,\aX[,\(��b =X][,\[,] = , = X[,\(�� and AX[,\ ∘D X[,\B(�� = AX][,\ ∘ X[,\B(�� = X][,\aX[,\(��b =X][,\[2] = 2 = X[,\(��. This shows that X[,\ is idempotent.
Proposition 5.3 X�j is idempotent.
Proof. Since X�j is an identity in YZT9��A(��, (2�B and by Lemma 4.7, we obtain that X]�j[,] = , and X]�j[2] = 2. By Lemma 5.2, we have that X�j is idempotent.
Theorem 5.4 Let X[,\ ∈ YZT9��A(��, (2�B. Then the following statements hold. (i) Every X[,\ ∈ q' is idempotent.
(ii) Every X[,\ ∈ qt is idempotent.
(iii) Every X[,\ ∈ qu is not idempotent.
Proof. We first prove that X[,\ ∈ q' is idempotent. To do this, let X[,\ ∈ q' with , = &�
and 2 = &c('� ≈ &c(0�. We consider X][,\[&�] = &� and X][,\a&c('� ≈ &c(0�b =X][,\a&c('�b ≈ X][,\a&c(0�b = &c('� ≈ &c(0�. By Lemma 5.2, X[,\ ∈ q' is idempotent.
Next, let X[,\ ∈ qt with , = &� and 2 = ¬(&c('� ≈ &c(0��. We consider X][,\[&�] = &�
and X][,\a¬(&c('� ≈ &c(0��b = ¬AX][,\a&c('� ≈ &c(0�bB = ¬A&c('� ≈ &c(0�B and then by Lemma 5.2, X[,\ ∈ qt is idempotent. Lastly, let X[,\ ∈ qu with , = &� and 2 =¬�(&c('�, &c(0��. To show that it is not idempotent, we consider X][,\a¬�A&c('�, &c(0�Bb = ¬AX][,\a�A&c('�, &c(0�BbB = ¬RP09��0AX[,\(��, X][,\a&c('�b, X][,\a&c(0�bBS = ¬RP09��0A¬�A&c('�, &c(0�B, &c('�, &c(0�BS
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= ¬�¬ RP09��0A�A&c('�, &c(0�B, &c('�, &c(0�BS� = � R&cAc('�B, &cAc(0�BS ≠ ¬�A&c('�, &c(0�B. Therefore, every X[,\ ∈ qu is not idempotent.
The following example shows that there is an element in q0 which is not idempotent.
Example 5.5 Let ((3),(2)) be a type, i.e., we have one ternary operation symbol and
one binary relation symbol, say � and �, respectively. If we consider X[,\ ∈ q0 with
, = &0 and 2 = �(&0, &'�, then we obtain X][,\[�(&0, &'�] = P09��0AX[,\(��, &0, &'B =P09��0(�(&0, &'�, &0, &'� = �(&', &0� ≠ �(&0, &'�. So, X[,\ in this form is not idempotent.
We have to find some necessary conditions for the element in q0 which is idempotent element. The next theorem shows such condition.
Theorem 5.6 Let X[,\ ∈ q0. Then X[,\ is idempotent if and only if ^( � = for all = 1,2. Proof. Let X[,\ ∈ q0. Then we have , = &� and 2 = �A&c('�, &c(0�B. Assume that ^( � ≠ for some = 1,2. We prove that X[,\ is not idempotent. To show this, we
considerX][,\a�A&c('�, &c(0�Bb = X][,\[�(&0, &'�] = P09��0 (�(&0, &'�, &0, &'� = �(&', &0� ≠ �(&0, &'� and then by Lemma 5.2, X[,\ is not idempotent. Conversely, assume that the
condition holds. Clearly, X][,\[&�] = &� and we see that X][,\a�A&c('�, &c(0�Bb =X][,\[�(&', &0�] = P09��0(�(&', &0�, &', &0� = �(&', &0� and thus by Lemma 5.2 we get that X[,\ is idempotent.
Now, it comes to characterize the idempotent element in qv, … , qy. We first show that all elements in qy are not idempotent and then show that the idempotency of qv, qw, qx need the some conditions. In fact, we have the following results.
Theorem 5.7 Every X[,\ ∈ qy is not idempotent.
Proof. Let X[,\ ∈ qy with , = �A&N('�, … , &N(��B, 2 = ¬�(&c('�, &c(0��. Suppose the contrary that X[,\ is idempotent, by Lemma 5.2, we obtain that X][,\[,] = , and X][,\[2] = 2. Obviously, X][,\a¬�(&c('�, &c(0��b ≠ ¬�(&c('�, &c(0�� since we have
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already shown this inequality holds in Theorem 5.4(iii). It contradicts to the result of
our assumption. Therefore, X[,\ is not idempotent.
Next, we show that there is an element in qv which is not idempotent as the following example.
Example 5.8 Let ((3),(2)) be a type, i.e., we have one ternary operation symbol and
one binary relation symbol, say � and �, respectively. If we consider X[,\ ∈ qv with
, = �(&t, &', &0� and 2 = &' ≈ &0., then we have
X][,\[�(&t, &', &0�] = Jt9��t(�(&t, &', &0�, &t, &', &0� = �(&0, &t, &'�. By Lemma 5.2, we conclude that X[,\ is not idempotent.
We remark here that if we let X[,\ ∈ qv,…,qy, then X][,\[2] has the same situation in the previous theorems. So, we are interesting in the way to find some
conditions for the idempotency of X][,\[,]. The next theorem shows that if we set some conditions, then we get the characterization of idempotent elements in qv, qw, qx.
Theorem 5.9 Let X[,\ ∈ YZT9��A(��, (2�B. Then the following statements hold. (i) X[,\ ∈ qv is idempotent if and only if O(�� = � for all � = 1,… , �. (ii) X[,\ ∈ qw is idempotent if and only if O(�� = � for all � = 1,… , � and
^( � = for all = 1,2. (iii) X[,\ ∈ qx is idempotent if and only if O(�� = � for all � = 1,… , �.
Proof. (i) Let X[,\ ∈ qv with , = �A&N('�, … , &N(��Band2 = &c('� ≈ &c(0�. Now we
may assume that if O(�� ≠ � for some � = 1,… , �. Then X][,\a�A&N('�, … , &N(��Bb =J�9���A�A&N('�, … , &N(��B, &N('�, … , &N(��B = �(&NAN('�B, … , &NAN(��B�. By our assumption, �(&NAN('�B, … , &NAN(��B� ≠ �A&N('�, … , &N(��B and thus X[,\ is not
idempotent. Conversely, assume that the condition holds. To show that X[,\ is
idempotent we consider X][,\a�A&N('�, … , &N(��Bb = X][,\[�(&', … , &��] = �(&', … , &�� so that X][,\[,] = ,. We can prove similarly to the proof of Theorem 5.4(i) that X][,\[2] =2. Therefore, X[,\ is idempotent.
(ii) Let X[,\ ∈ qw with = �A&N('�, … , &N(��Band 2 = �(&c('�, &c(0��. We first assume that O(�� ≠ � for some � = 1,… , � or ^( � ≠ for some = 1,2. Then by the same manner as in the proof of (i) we can show that X[,\ is not idempotent. Conversely,
assume that the condition holds. Clearly,
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X][,\a�A&N('�, … , &N(��Bb = X][,\[�(&', … , &��] = �(&', … , &�� and thus X][,\[,] = ,. Moreover, we have that X][,\a�A&c('�, &c(0�Bb = X][,\[�(&', &0�] = �(&', &0�, that is X][,\[2] = 2. By Lemma 5.2, X[,\ is idempotent.
(iii) By using Lemma 5.1, we can prove similarly to the proof of (i) that this
statement holds.
Note that every idempotent element is regular. We characterize all regular
elements in YZT9��A(��, (2�B, we consider X[,\ ∈ YZT9��A(��, (2�B which is not idempotent. The characterization of regularity in YZT9��A(��, (2�B can be shown in the next theorem.
Theorem 5.10 Let X[,\ ∈ YZT9��A(��, (2�B. Then the following statements hold. (i) Every X[,\ ∈ q0 is regular. (iv) Every X[,\ ∈ qw is regular.
(ii) Every X[,\ ∈ qu is regular. (v) Every X[,\ ∈ qx is regular.
(iii) Every X[,\ ∈ qv is regular. (vi) Every X[,\ ∈ qy is regular.
Proof. (i) Let X[,\ ∈ q0 with , = &� and 2 = �(&c('�, &c(0��. We consider regularity of X[,\ ∈ q0 only the case of ^( � ≠ for some = 1,2. To do this, we choose X[{ ,\{ ∈ q0
with ,́ = &� and 2{ = �(&c�d('�, &c�d(0�� such that AX[,\ ∘D X[{ ,\{ ∘D X[,\B(�� = &� =X[,\(�� and AX[,\ ∘D X[{ ,\{ ∘D X[,\B(�� = X][,\aX][{ ,\{ [�(&c('�, &c(0��]b =X][,\aP09��0AX[{ ,\{ (��, &c('�, &c(0�Bb = X][,\aP09��0A�(&c�d('�, &c�d(0��, &c('�, &c(0�Bb =X][,\ ��(&cRc�d('�S, &cRc�d(0�S�� = X][,\ g�(&Ac∘c�dB('�, &Ac∘c�dB(0��h = X][,\[�(&', &0�] =P09��0A�(&c('�, &c(0��, &', &0B = �A&c('�, &c(0�B = X[,\(��. This implies that, X[,\ is
regular.
(ii) Similarly to the proof of (i) and by using Lemma 5.1, we can show that
every X[,\ ∈ qu is regular.
(iii) Let X[,\ ∈ qv with , = �A&N('�, … , &N(��Band2 = &c('� ≈ &c(0�. We
consider in the case of O(�� ≠ � for some � = 1,… , �, then there exists X[{ ,\{ ∈ q0 with
(iv) Let X[,\ ∈ qw with , = �A&N('�, … , &N(��Band2 = �A&c('�, &c(0�B. To prove that X[,\ is regular, we consider in three cases: If O(�� = � for all � = 1,… , � and ^( � ≠ for some = 1,2, then there exists X[{ ,\{ ∈ qw with ,́ = �A&N('�, … , &N(��B and 2 ={ �(&c�d('�, &c�d(0�� such that AX[,\ ∘D X[{ ,\{ ∘D X[,\B(��
Similarly to the proof of (i), we have that AX[,\ ∘D X[{ ,\{ ∘D X[,\B(�� = X[,\(��. If O(�� ≠ � for some � = 1,… , � and ^( � = for all = 1,2, then there exists X[{ ,\{ ∈ qw
with ,́ = �A&N�d('�, … , &N�d(��B and 2 ={ �A&c('�, &c(0�B such that AX[,\ ∘D X[{ ,\{ ∘D X[,\B(�� = X[,\(��, it follows from (iii). Moreover, we consider AX[,\ ∘D X[{ ,\{ ∘D X[,\B(��
= X][,\aX][{ ,\{ [�A&c('�, &c(0�B]b
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Finally, if O(�� ≠ � for some � = 1,… , � and ^( � ≠ for some = 1,2, then there exists X[{ ,\{ ∈ qw with ,́ = �A&N�d('�, … , &N�d(��B and 2 ={ �(&c�d('�, &c�d(0�� such that AX[,\ ∘D X[{ ,\{ ∘D X[,\B(�� = X[,\(�� and AX[,\ ∘D X[{ ,\{ ∘D X[,\B(�� = X[,\(��. Therefore, we conclude that X[,\ is regular.
(v) This statement can be proved by using Lemma 5.1 and the same process as
we proved in (iii).
(vi) This statement can be proved by using Lemma 5.1 and the same process as
we proved in (iv).
Consequence of this section, every linear hypersubstitution is regular and then
YZT9��A(��, (2�B is a regular semigroup.
Acknowledgements This research was supported by the Centre of Excellence in
Mathematics, the Commission on Higher Education, Thailand.
References
Burris, S., & Sankappanavar, H. P. (1981). A Course in Universal Algebra. New York:
Springer Verlag.
Couceiro, M., & Lehtonen, E. (2012). Galois Theory for sets of operations closed inder
permutation, cylindrification and composition. Algebra universalis, 67, 273-
297.
Denecke, K. (2016). The Partial Clone of Linear Terms. Siberian Mathematical
Journal, 57,589–598.
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