Linear Cohypersubstitutions and Generalized Linear Cohypersubstitutions of type \tau = (n)

LINEAR COHYPERSUBSTITUTIONS AND GENERALIZED LINEAR COHYPERSUBSTITUTIONS OF TYPE τ = (n)

JULALUK BOONSOL

A Thesis submitted in partial fulfillment of the requirements for the degree of Master of Science in Mathematics

at Mahasarakham University October 2019

All rights reserved by Mahasarakham University

LINEAR COHYPERSUBSTITUTIONS AND GENERALIZED LINEAR COHYPERSUBSTITUTIONS OF TYPE τ = (n)

JULALUK BOONSOL

A Thesis submitted in partial fulfillment of the requirements for the degree of Master of Science in Mathematics

at Mahasarakham University October 2019

All rights reserved by Mahasarakham University

The examining committee has unanimously approved this thesis, submitted by Miss Julaluk Boonsol, as a partial fulfillment of the requirements for the Master of Science in Mathematics at Mahasarakham University.

Examining Committee

. . . Chairman

(Asst. Prof. Somsak Lekkoksung, Dr.rer.nat.) (External expert)

. . . Committee (Asst. Prof. Kittisak Saengsura, Dr.rer.nat.) (Advisor)

. . . Committee

(Asst. Prof. Ananya Anantayasethi, Dr.rer.nat.) (Faculty graduate committee)

. . . Committee

(Asst. Prof. Jeeranunt Khampakdee, Ph.D.) (Faculty graduate committee)

Mahasarakham University has granted approval to accept this thesis as a partial fulfillment of the requirements for the Master of Science in Mathematics.

. . . . (Prof. Pairot Pramual, Ph.D.) (Asst. Prof. Krit Chaimoon, Ph.D.) Dean of the Faculty of Science Dean of Graduate School

. . . 2019

ACKNOWLEDGEMENTS

It would be difficult to name all to whom I owe thanks for their assistance and help with this work. However special acknowledgement is extended to the following:

I gratefully acknowledge the financial support of the Science Achievement Scholarship of Thailand (SAST).

I express my gratitude to my advisor, Asst. Prof. Kittisak Saengsura, who introduced me to research, for his help and constant encouragement throughout the course of this research. I am the most grateful for his teaching and advice. This thesis would not have been completed without all the support that I have always received from him.

I would like to thanks Asst. Prof. Somsak Lekkoksung, Rajamangala University of Technology Isan Khonkaen Campus Khonkaen Thailand, Asst. Prof. Ananya Anantayasethi, and Asst. Prof. Jeeranunt Khampakdee, Department of Mathematics, Faculty of Science, Mahasarakham University for their kind comments and support as members of dissertation committee.

I express my family for their understanding and support me for everything until this study completion.

Finally, thanks are due to all of the staffs in the Science Achievement Scholarship of Thailand (SAST) and Department of Mathematics, Faculty of Science, Mahasarakham University, for their warm welcome.

Julaluk Boonsol

i

ii

TITLE Linear Cohypersubstitutions and Generalized Linear Cohypersubstitutions of type τ = (n)

CANDIDATE Miss Julaluk Boonsol

DEGREE Master of Science MAJOR Mathematics ADVISOR Asst. Prof. Kittisak Saengsura, Dr.rer.nat.

UNIVERSITY Mahasarakham University YEAR 2019

ABSTRACT

A coterm of type τ was introduced by K. Denecke and K. Saengsura in 2009 as a compound of two disjoint sets {f_i |i∈I} and {eⁿ_i |n ∈N,0≤ i≤n−1}. A linear coterm t is a coterm which the element of {eⁿ_i | n ∈ N,0 ≤ i ≤ n−1} occurring in t are all different. In this research, we study some semigroup properties of linear cohypersubstitutions and generalized linear cohypersubstitutions of type τ = (n), for instance: idempotents, regulars and Green’s relations.

Keywords: linear cohypersubstitution, generalized linear cohypersubstitution, idempotent, regular, Green’s relations.

CONTENTS

Page

Acknowledgements i

Abstract ii

Contents iii

Chapter 1 Introduction 1

1.1 Background 1

Chapter 2 Preliminaries 4

2.1 Semigroups 4

2.2 Green’s relations 5

2.3 Algebras 6

2.4 Coalgebras 9

Chapter 3 Linear cohypersubstitutions of type τ = (n) 13 3.1 Monoid of linear cohypersubstitutions of type τ = (n) 13 3.2 Idempotent and some regular elements of linear cohypersubstitutions

of type τ = (n) 17

3.3 Green’s relations of linear cohypersubstitutions of type

τ = (n) 22

Chapter 4 Generalized linear cohypersubstitutions of type τ = (n) 29 4.1 Monoid of generalized linear cohypersubstitutions of

type τ = (n) 29

4.2 Idempotent and some regular elements of generalized linear cohypersubstitutions of type τ = (n) 41 4.3 Green’s relations of generalized linear cohypersubstitutions

of type τ = (n) 45

Chapter 5 Conclusions 51

References 58

Biography 61

iii

CHAPTER 1 INTRODUCTION

1.1 Background

In this dissertation, we mainly pay attention to the concept of cohypersubstitutions and generalized cohypersubstitutions.

Let A be a non-empty set and n be a positive integer. The n-th copower A^tn of A is the union of n disjoint copies of A; formally, we define A^tn as the cartesian product A^tn :=n×A, where n :={0, ..., n−1}. An element (i, a) in this copower corresponds to the element a in the i-th copy of A, for 1≤i≤n. A co-operation on A is a mapping f^A : A → A^tn for some n ≥ 1; the natural number n is called the arity of the co-operation f^A. We also need to recall that any n-ary co-operation f^A on set A can be uniquely expressed as a pair (f₁^A, f₂^A) of mappings, f₁^A :A→ n and f₂^A : A → A; the first mapping gives the labelling used by f^A in mapping elements to copies of A, and the second mapping tells us what element of A any element is mapped to.

We shall denote by cO⁽ⁿ⁾_A = {f^A|A → A^tn} the set of all n-ary co-operations defined on A, and by cO_A := ∪n≥1cO_A⁽ⁿ⁾ the set of all finitary co-operations defined on A. An indexed coalgebra is a pair(A; (f_i^A)i∈I), wheref_i^A is an n_i-ary co-operation defined onA,andτ = (n_i)_i∈I forn_i ≥1is called the type of the coalgebra. Coalgebras were studied by Drbohlav [10]. In [1], the following superposition of co-operations was introduced. If f^A ∈ cO⁽ⁿ⁾_A and g^A₁, ..., g_n^A ∈ cO_A^(k), then the k-ary co-operation f^A[g₁^A, ..., g^A_n] :A→A^tk is defined by

a7→((g^A_fA

1(a))₁(f₂^A(a)),(g_f^AA

1(a))₂(f₂^A(a))),

for all a ∈ A. The co-operation f^A[g^A₁, ..., g_n^A] is called the superposition of f^A and g₁^A, ..., g_n^A. It will also be denoted by compⁿ_k(f^A, g₁^A, ..., g^A_n).

1

2 The injection co-operations ι^n,A_i :A → A^tn are special co-operations which are defined for each 0≤i≤n−1 by ι^n,A_i :A→A^tn with a 7→(i, a) for all a ∈A. Then we get a many-sorted algebra

((cO_A⁽ⁿ⁾)n≥1,(compⁿ_k)k,n≥1,(ι^n,A_i )0≤i≤n−1),

called the clone of co-operations on A. In [1], it is mentioned that this algebra is a clone, i.e., it satisfies the three clone axioms (C1),(C2),(C3). In [8], K. Denecke and K. Saengsura gave a full proof of this fact.

In 2016, D. Boonchari and K. Saengsura studied the monoid of cohypersubstitutions of type τ = (n) and characterized all idempotent and regular elements of set of all cohypersubstitutions (see [2]). The concept of cohypersubstitutions can be extended into the concept of generalized cohypersubstitutions defined by S. Jermjitpornchai and N. Saengsura in 2013 which we could find an original literature in [12]. Similar to the original concept of cohypersubstitutions, the set of all generalized cohypersubstitutions together with a special binary operation forms a monoid.

For our purposes, we introduce the notions of linear cohypersubstitutions of type τ = (n), then idempotent, some regular elements and Green’s relations of linear cohypersubstitutions of type τ = (n) are studied. Moreover, we study some properties of generalized linear cohypersubstitutions of type τ = (n). We devide our thesis into 5 chapters, as follows:

In the first chapter, the introduction was presented.

In chapter 2, we present some basic concepts and results of algebras and coalgebras without proof which are needed in the subsequent chapters.

In chapter 3, we introduce the concept of linear cohypersubstitutions of type τ = (n). We also study the properties of idempotent, regular elements and Green’s relations of monoid of linear cohypersubstitutions of type τ = (n).

3

In chapter 4, we introduce the concept of generalized linear cohypersubstitutions of type τ = (n). We also study the properties of idempotent, regular elements and Green’s relations of monoid of generalized linear cohypersubstitutions of typeτ = (n).

In the last chapter, we summarize results of our study.

CHAPTER 2 PRELIMINARIES

In this chapter, we will give some definitions, notations, dealing with some preliminaries and some useful results that will be duplicated in later chapters.

2.1 Semigroups

Definition 2.1.1. [4] Let S be a non-empty set, and ∗ said to be a binary operation on S, if a∗b is defined for all a, b∈S.

Definition 2.1.2. [4] A groupoid (S,∗) is defined as a non-empty set S on which a binary operation ∗, by which we mean a map ∗:S×S → S is defined. We say that (S,∗) is a semigroup if the operation ∗ is associative, that is to say, if for all x, y and z in S,

((x∗y)∗z) = (x∗(y∗z)).

Definition 2.1.3. [4] If a semigroup (S,∗) contains an element 1 with the property that, for all x in S,

x∗1 = 1∗x=x,

we say that 1 is an identity element (or just an identity) of S, and that (S,∗) is a semigroup with identity or (more usually) a monoid.

Definition 2.1.4. [4] An element a of a semigroup (S,∗) is called idempotent if a∗a=a.

Definition 2.1.5. [4] An element a of a semigroup (S,∗) is called regular if there exists x in S such that a∗x∗a =a. The semigroup (S,∗) is called regular if all its elements are regular.

4

5 2.2 Green’s relations

Definition 2.2.1. [4] For semigroup S, we define

S¹ :=







S if S has an identity element S∪ {1} otherwise.

Definition 2.2.2. [4] If a is an element of a semigroup S without identity then Sa need not contain a. The following notations will be standard:

S¹a =Sa∪ {a}

aS¹ =aS∪ {a}

S¹aS¹ =SaS∪Sa∪aS ∪ {a}.

Notice that S¹a, aS¹ and S¹aS¹ are all subsets of S-they do not contain the element 1.

Remark. We call S¹a, aS¹ and S¹aS¹ the principal left ideal generated by a.

Definition 2.2.3. [4] An equivalence L on S is defined by the rule that aLb if and only ifa andb generate the same principal left ideal, that is, if and only if S¹a =S¹b.

Similarly, we define the equivalence R by the rule that aRb if and only if aS¹ =bS¹. Proposition 2.2.4. [4] Let a, b be elements of a semigroup S. Then aLb if and only if there exist x, y in S¹ such that xa=b, yb=a. Also, aRb if and only if there exist u, v in S¹ such that au=b, bv =a.

Definition 2.2.5. [4] Let a, b be elements of a semigroup S. An equivalence H on S is defined by the rule that aHb if and only if aLb and aRb.

Definition 2.2.6. [4] Let a, b be elements of a semigroup S. An equivalence D on S is defined by the rule that aDb if and only if there exists c in S such that aLc and cRb.

Definition 2.2.7. [4] An equivalence J on S is defined by the rule that aJb if and only if S¹aS¹ =S¹bS¹, that is to say, if and only if there exist x, y, u, v in S¹ such that xay =b, ubv =a.

6 2.3 Algebras

Denote the set of all positive integers by N⁺. That is, N⁺:={1,2,3, ...}.

Definition 2.3.1. [9] An algebra is a pair A := (A;F^A) consisting of a non-empty set A with the set F^A = {f_i^A|i ∈ I} of operations defined on A indexed by some non-empty set I, where for each i∈I, f_i^A is an n_i-ary operation defined on A.

Let A:= (A;F^A) be an algebra. The set A is called the universe of the algebra, and the f_i^A are the basic operations where i ∈ I. Moreover, a non-negative integer n_i is called the arity of the basic operation f_i^A, and the sequence of all arities τ := (n_i)i∈I is called the similarity type, for short type, of the algebra A. However, we may consider F^A as (f_i^A)i∈I for some index set I. For convenience, we may write (A; (f_i^A)i∈I) instead of(A;F^A). Denoted byAlg(τ)the class of all algebras of a given type τ.

Let X_n :={x₁, ..., x_n}, where n∈N⁺. The element x_i in X_n is called variable.

Definition 2.3.2. [9] An n-ary term of type τ is defined in the following inductive ways.

(1) Every variable in X_n is an n-ary term.

(2) If t₁, ..., t_ni are n-ary terms and f_i is n_i-ary operation symbol, then f_i(t₁, ..., t_ni) is an n-ary term.

We denote the set of all n-ary terms of type τ by W_τ(X_n). The set W_τ(X_n) is the smallest set containing X_n and is closed under finite application of (2). Denoted by W_τ(X) the set of all terms of type τ. That is,

W_τ(X) := [

n∈N⁺

W_τ(X_n).

Definition 2.3.3. [3] Let m, n∈ N⁺. The operation S_n^m :W_τ(X_m)×(W_τ(X_n))^m → W_τ(X_n) is defined inductively as follows:

(1) S_n^m(x_i, t₁, ..., t_m) := t_i, where 1≤i≤m.

7 (2) S_n^m(f_i(s₁, ..., s_ni), t₁, ..., t_m) := f_i(S_n^m(s₁, t₁, ..., t_m), ..., S_n^m(s_ni, t₁, ..., t_m)) and

assume that S_n^m(sj, t1, ..., tm) is already defined for all 1≤j ≤ni.

Definition 2.3.4. [6] A hypersubstitution is a mapping from the set of all operation symbols to the set of terms preserving arity. Denoted by Hyp(τ) the set of all hypersubstitutions of type τ.

Any hypersubstitution σ can be extened to mapping σˆ :Wτ(X)→Wτ(X) defined by the following steps:

(1) σ[x] :=ˆ x for all variable x∈X.

(2) σ[fˆ _i(t₁, ..., t_ni)] := S_nⁿⁱ(σ(f_i),σ[tˆ ₁], ...,σ[tˆ _ni]) if f_i(t₁, ..., t_ni) ∈ W_τ(X_n) and ˆ

σ[tj]∈Wτ(Xn) is already defined for all 1≤j ≤ni.

In [6], the author showed that the set Hyp(τ) of all hypersubstitutions of type τ endowed with a binary operation ◦_h, defined by σ₁◦_hσ₂ := ˆσ₁◦σ₂, forms a monoid, where σ_id, defined by σ_id(f_i) := f_i(x₁, ..., x_ni), is its neutral element. The algebraic system Hyp(τ) := (Hyp(τ);◦_h, σ_id) is a monoid.

Definition 2.3.5. [11] Let n ∈ N. An (n+ 1)-ary generalized superposition is an (n+ 1)-ary operation Sⁿ on the set Wτ(X) defined by the following steps:

(1) if t =xi ∈Xn, then Sⁿ(t, t1, ..., tn) :=ti, (2) if t =x_j ∈X\X_n, then Sⁿ(t, t₁, ..., t_n) := x_j,

(3) if t = f_i(s₁, ..., s_ni) and assume that Sⁿ(s_j, t₁, ..., t_n) is already defined for all 1≤j ≤n, then

Sⁿ(t, t1, ..., tn) :=fi(Sⁿ(s1, t1, ..., tn), ..., Sⁿ(sni, t1, ..., tn)).

Then we define the extension σˆ :W_τ(X)→W_τ(X) by the following steps:

(1) If t ∈X, then σ[t] :=ˆ t.

(2) Ift=f_i(t₁, ..., t_ni),thenσ[t] :=ˆ Sⁿⁱ(σ(f_i),σ[tˆ ₁], ...,σ[tˆ _ni]),whereσ[tˆ _j]is already defined for all 1≤j ≤n_i.

8 This extension is uniquely determined and allows us to define a multiplication, denoted by ◦G, on the set HypG(τ) of all generalized hypersubstitions of type τ by

σ₁◦_Gσ₂ := ˆσ₁◦σ₂

where ◦ denotes the usual composition of mappings. Let σ_id be the identity hypersubstitution which maps eachn_i-ary operation symbolf_i to the termf_i(x₁, ..., x_ni).

The algebraic properties Hyp_G(τ) := (Hyp_G(τ);◦_G, σ_id) is a monoid.

Let var(t) be the set of variables occurring in t.

Definition 2.3.6. [5] Ann-ary linear term of typeτ is defined by induction as follows:

(i) x_j ∈X_n is an n-ary linear term of type τ for all j ∈ {1, ..., n}.

(ii) If t₁, ..., t_ni are n-ary linear terms of type τ and var(t_j)∩var(t_k) = ∅ for all 1≤j < k ≤n_i, then f_i(t₁, ..., t_ni) is an n-ary linear term of type τ.

(iii) The set W_τ^lin(X_n) of all n-ary linear terms of type τ is the least set containing x₁, ..., x_n and closed under finite applications of (ii).

W_τ^lin(X) denotes the union of all W_τ^lin(X_n) for n∈N⁺.

Definition 2.3.7. [5] The partial many-sorted mappingS^m_n :W_τ^lin(X_m)×(W_τ^lin(X_n))^m (→W_τ^lin(X_n) by

S^m_n(t, s₁, ..., s_m) :=







S_n^m(t, s1, ..., sm) if var(sj)∩var(sk) = ∅ for all 1≤j < k≤m, not defined otherwise

for m, n∈N⁺.

Definition 2.3.8. [5] A linear hypersubstitution is a mapping from the set of all operation symbols to the set of linear terms preserving arity. Denoted by Hyp^lin(τ) the set of all linear hypersubstitutions of type τ.

Any linear hypersubstitutionαcan be extened to mapping αˆ:W_τ^lin(X)→W_τ^lin(X) defined by the following steps:

(1) α[x] :=ˆ x for all variable x∈X.

9 (2) α[fˆ _i(t₁, ..., t_ni)] := Sⁿ_nⁱ(α(f_i),α[tˆ ₁], ...,α[tˆ _ni]) if f_i(t₁, ..., t_ni) ∈ W_τ^lin(X_n) and

ˆ

α[tj] is already defined for all 1≤j ≤ni.

In [5], the author showed that the set Hyp^lin(τ) of all linear hypersubstitutions of type τ endowed with a binary operation ◦_h, defined by α₁ ◦_hα₂ := ˆα₁◦α₂, forms a monoid, where α_id, defined by α_id(f_i) := f_i(x₁, ..., x_ni), is its neutral element. The algebraic properties Hyp^lin(τ) := (Hyp^lin(τ);◦h, αid) is a monoid.

2.4 Coalgebras

Denote the set of all positive integers by N⁺. That is, N⁺:={1,2,3, ...}.

Definition 2.4.1. [8] A coalgebra of type τ is a system (A; (f_i^A)i∈I) consisting of a non-empty set A and a set of finitary co-operations f_i^A : A → A^tni, where n_i is the arity of the co-operation f_i^A and A^tni is the n_i-th copower of A. We recall that such copowers are defined by A^tni := {1, ..., ni} ×A. This means that each ni-ary co-operation f_i^A is uniquely determined by a pair ((f_i^A)₁,(f_i^A)₂) of mappings, (f_i^A)₁ from A to {1, ..., n_i} and (f_i^A)₂ from A to A.

Definition 2.4.2. [8] Let τ = (n_i)i∈I be a type of coalgebra, that is, an indexed set of natural numbers, ni ≥ 1, for all i ∈ I, with corresponding indexed set (fi)i∈I of co-operation symbols. We say that symbol f_i has arity n_i, for i ∈ I. Let S

{eⁿ_j | n ≥ 1, n ∈ N,0 ≤ j ≤ n−1} be a set of symbols which is disjoint from the set {f_i |i∈I}. We assign to each eⁿ_j the positive integer n as its arity. Then coterms of type τ are defined as follows:

1. For every i∈I the co-operation symbol f_i is an n_i-ary coterm of type τ.

2. For every n≥1 and 0≤j ≤n−1 the symbol eⁿ_j is an n-ary coterm of type τ.

3. If t₁, ..., t_ni are n-ary coterms of type τ, then f_i[t₁, ..., t_ni] is an n-ary coterm of type τ and if t₀, ..., t_n−1 are m-ary coterms of type τ, then eⁿ_j[t₀, ..., t_n−1] is an m-ary coterm of type τ, for every i∈I and n≥1 and 0≤j ≤n−1.

Let cTτ⁽ⁿ⁾ be the set of all n-ary coterms of type τ and let cT_τ := [

n≥1

cT_τ⁽ⁿ⁾ be the set of all (finitary) coterms of type τ.

10 Definition 2.4.3. [7] The operation S_mⁿ : cTτ⁽ⁿ⁾ ×(cTτ^(m))ⁿ → cTτ^(m) is defined by induction on the complexity of coterm, as follows:

(i) S_mⁿ(eⁿ_i, t₀, ..., t_n−1) :=t_i for 0≤i≤n−1.

(ii) S_nⁿⁱ

i(f_i, eⁿ₀ⁱ, ..., eⁿ_nⁱ

i−1) := f_i for an n_i-ary co-operation symbol f_i.

(iii) Sm^nj(g_j, t₁, ..., t_nj) :=g_j[t₁, ..., t_nj] if g_j is an n_j-ary co-operation symbol.

(iv) S_mⁿ(fi[s1, ..., sni], t1, ..., tn) := fi[S_mⁿ(s1, t1, ..., tn), ..., S_mⁿ(sni, t1, ..., tn)] where fi

is an n_i-ary co-operation symbol, s₁, ..., s_ni are n-ary coterms of type τ and t₁, ..., t_n are m-ary coterms of type τ.

These operations give us a many-sorted algebra

cT_τ := ((cT_τ⁽ⁿ⁾)_n≥1,(S_mⁿ)_m,n≥1,(eⁿ_j)_{0≤j≤n−1}).

Theorem 2.4.4. [7] The many-sorted algebra cT_τ satisfies the following identities:

(C1) Sˆ_m^p(z,Sˆ_mⁿ(y₁, x₁, ..., x_n), ...,Sˆ_mⁿ(y_p, x₁, ..., x_n))≈ Sˆ_mⁿ( ˆS_n^p(z, y₁, ..., y_p), x₁, ..., x_n), (m, n, p∈N⁺),

(C2) Sˆ_mⁿ(eⁿ_i, x₀, ..., xn−1)≈x_i (m ∈N⁺,0≤i≤n−1), (C3) Sˆ_nⁿ(y, eⁿ₀, ..., eⁿ_n−1)≈y, (n∈N⁺).

(here Sˆ_mⁿ, eⁿ_i are operation symbols corresponding to the clone type.)

Let (A; (f_i^A)i∈I) be a coalgebra of type τ. Then each coterm of type τ induces a cooperation which is inductively defined as follows:

(i) If f_i is an n_i-ary cooperation symbol, then f_i^A is the induced n_i-ary cooperation.

(ii) (eⁿ_j)^A:=ι^n,A_j for every n≥1,0≤j ≤n−1, where ι^n,A_j is an n-ary injection.

(iii) If fi[g1, ..., gni] is a coterm and if we inductively assume that the induced cooperations g₁^A, ..., g^A_ni are known, then (f_i[g₁, ..., g_ni])^A=f_i^A[g^A₁, ..., g_n^A_i].

(iv) If g₀^A, ..., g_n−1^A are assumed to be known, then we define (eⁿ_j[g₀, ..., gn−1])^A =g^A_j for 0≤j ≤n−1.

11 Definition 2.4.5. [7] A cohypersubstitution of type τ is a mapping σ:{f_i |i∈I} → cTτ from the set of all co-operation symbols to the set of all coterms which preserves the arities. The extension of σ is a mapping σˆ : cT_τ → cT_τ which is inductively defined by the following steps:

(i) σ[eˆ ⁿ_j] :=eⁿ_j for every n ≥1 and 0≤j ≤n−1, (ii) σ[fˆ i] :=σ(fi) for every i∈I,

(iii) σ[fˆ _i[t₁, ..., t_ni]] :=S_nⁿⁱ(σ(f_i),σ[tˆ ₁], ...,σ[tˆ _ni]) for t₁, ..., t_ni ∈cTτ⁽ⁿ⁾.

Let Cohyp(τ) be the set of all cohypersubstitutions of type τ. On the set Cohyp(τ) of all cohypersubstitutions of type τ we may define a binary operation

◦coh by σ1◦cohσ2 := ˆσ1◦σ2 where ◦ is the usual composition of mappings. Let σid

be the cohypersubstitution defined by σ_id(f_i) := f_i for all i ∈ I. Then we have that (Cohyp(τ);◦_coh, σ_id)forms a monoid which is called the monoid of cohypersubstitution of type τ where the cohypersubstitution σ_id satisfies the equation σˆ_id[t] = t for all t∈cT_τ.

Definition 2.4.6. [12] Let m ∈ N⁺, a generalized superposition of coterms: S^m : (cT_τ)^(m+1) →cT_τ defined inductively by the following steps:

(i) If t =eⁿ_i and 0≤i≤m−1, then

S^m(eⁿ_i, t₀, ..., tm−1) = t_i, where t₀, ..., tm−1 ∈cT_τ. (ii) If t =eⁿ_i and 0< m≤i≤n−1, then

S^m(eⁿ_i, t₀, ..., tm−1) = eⁿ_i, where t₀, ..., tm−1 ∈cT_τ. (iii) If t =f_i[s₁, ..., s_ni], then

S^m(t, t₁, ..., t_m) =f_i(S^m(s₁, t₁, ..., t_m), ..., S^m(s_ni, t₁, ..., t_m)) where S^m(s₁, t₁, ..., t_m), ..., S^m(s_ni, t₁, ..., t_m)∈cT_τ.

The above definition can be written in the following forms:

(i) If t =eⁿ_i and 0≤i≤m−1, then

eⁿ_i[t₀, ..., tm−1] =t_i, where t₀, ..., tm−1 ∈cT_τ.

12 (ii) If t =eⁿ_i and 0< m≤i≤n−1, then

eⁿ_i[t0, ..., tm−1] =eⁿ_i, where t0, ..., tm−1 ∈cTτ. (iii) If t =f_i[s₁, ..., s_ni], then

(f_i[s₁, ..., s_ni])[t₁, ..., t_m] =f_i[s₁[t₁, ..., t_m], ..., s_ni[t₁, ..., t_m]]

where s₁[t₁, ..., t_m], ..., s_ni[t₁, ..., t_m]∈cT_τ.

Definition 2.4.7. [12] A generalized cohypersubstitution of type τ is a mapping σ : {f_i|i ∈ I} → cT_τ. The extension of σ is a mapping σˆ : cT_τ → cT_τ which is inductively defined by the following steps:

(i) σ[eˆ ⁿ_j] :=eⁿ_j for every n ≥1 and 0≤j ≤n−1, (ii) σ[fˆ i] :=σ(fi) for every i∈I,

(iii) σ[fˆ _i[t₁, ..., t_ni]] :=Sⁿⁱ(σ(f_i),σ[tˆ ₁], ...,σ[tˆ _ni]) for t₁, ..., t_ni ∈cTτ⁽ⁿ⁾. Let Cohyp_G(τ) be the set of all generalized cohypersubstitutions of type τ.

In [12], on the set Cohyp_G(τ) of all generalized cohypersubstitutions of type τ we define a function◦CG:CohypG(τ)×CohypG(τ)→CohypG(τ)byσ1◦CGσ2 := ˆσ1◦σ2

for all σ₁, σ₂ ∈Cohyp_G(τ) where ◦ is the usual composition of mappings. Let σ_id be the generalized cohypersubstitution defined by σ_id(f_i) := f_i for all i∈I.

Proposition 2.4.8. [12] The set Cohyp_G(τ) of all generalized cohypersubstitutions of typeτ is associates with a binary operation ◦CGand the generalized cohypersubstitution σ_id is an identity of Cohyp_Gτ. We obtain a monoid (Cohyp_G(τ);◦_CG, σ_id).

CHAPTER 3

LINEAR COHYPERSUBSTITUTIONS OF TYPE τ = (n)

In this chapter, we will introduce the notions of linear cohypersubstitutions of type τ = (n).

3.1 Monoid of linear cohypersubstitutions of type τ = (n)

In this section, we will introduce the notions of the monoid of linear cohypersubstitutions.

Let t be a coterm of type τ. Then we denotes E(t) :={eⁿ_i |eⁿ_i occurring in t and 0≤i≤n−1}.

Definition 3.1.1. An n-ary linear coterm of type τ is defined by induction as follows:

(1) For every i∈I the co-operation symbol f is an n-ary linear coterm of type τ. (2) For every n ≥1 and 0≤j ≤n−1 the symbol eⁿ_j is an n-ary linear coterm of

type τ.

(3) If t₁, ..., t_n are n-ary linear coterms of type τ and E(t_j)∩E(t_k) = ∅ for all 1≤j < k ≤n, then f[t₁, ..., t_n] is an n-ary linear coterm of type τ.

We denote by cTτ^lin,(n) the set of all n-ary linear coterms of type τ and cT_τ^lin :=

[

n≥1

cT_τ^lin,(n) the set of all (finitary) linear coterms of type τ.

Example 3.1.2. Letτ = (2).Thenf[e³₀, e³₁], f[e³₁, e³₀]andf[f[e³₀, e¹₃], e³₂]are3-ary linear coterms of type τ.

Proposition 3.1.3. If t ∈cTτ^lin,(n), s₁, ..., s_n∈cTτ^lin,(m), and E(s_j)∩E(s_k) =∅ for all 1≤j < k≤n, then S_mⁿ(t, s₁, ..., s_n)∈cTτ^lin,(m).

Proof. We will prove by induction on the complexity of the linear coterm t.

(i) If t =eⁿ_i for 0≤i≤n−1, then

S_mⁿ(eⁿ_i, s₁, ..., s_n) =s_i+1 ∈cT_τ^lin,(m). 13

14 (ii) If t =f, then

S_nⁿ(f, s₁, ..., s_n) = S_mⁿ(f[eⁿ₀, ..., eⁿ_n−1], s₁, ..., s_n)

= f[sⁿ_m(eⁿ₀, s₁, ..., s_n), ..., sⁿ_m(eⁿ_n−1, s₁, ..., s_n)]

= f[s1, ..., sn]

∈ cT_τ^lin,(m).

(iii) If t = f[t₁, ..., t_n] ∈ cTτ^lin,(n), E(t_j)∩E(t_k) = ∅ for all 1 ≤ j < k ≤ n and assume that S_mⁿ(tj, s1, ..., sn)∈cTτ^lin,(m), then

S_mⁿ(f[t₁, ..., t_n], s₁, ..., s_n) = f[S_mⁿ(t₁, s₁, ..., s_n), ..., S_mⁿ(t_n, s₁, ..., s_n)]

∈ cT_τ^lin,(m).

Therefore, S_mⁿ(t, s₁, ..., s_n)∈cTτ^lin,(m).

Definition 3.1.4. The partial many-sorted mapping Sⁿ_m :cTτ^lin,(n)×(cTτ^lin,(m))ⁿ (→ cTτ^lin,(m) is defined by

Sⁿ_m(t, s1, ..., sn) :=







S_mⁿ(t, s₁, ..., s_n) if E(s_j)∩E(s_k) =∅ for 1≤j < k≤n, not defined otherwise.

For the extension of a cohypersubstitution σ the following holds:

Lemma 3.1.5. For any cohypersubstitution σ and any linear coterms t we have E(t)⊇E(ˆσ[t]).

Proof. Let σ ∈Cohyp(τ) and let t∈cTτ^lin,(n) be a linear coterm.

We will prove by induction on the complexity of the linear coterm t.

(1) If t =eⁿ_i where n≥1,0≤i≤n−1, then

E(t) = E(eⁿ_i) = E(ˆσ[eⁿ_i]) =E(ˆσ[t]).

(2) If t =f[eⁿ₀, ..., eⁿ_n−1], then

E(ˆσ[t]) = E(ˆσ[f[eⁿ₀, ..., eⁿ_n−1]])

15

= E(S_nⁿ(σ(f),σ[eˆ ⁿ₀], ...,σ[eˆ ⁿ_n−1]))

= E(S_nⁿ(σ(f), eⁿ₀, ..., eⁿ_n−1))

⊆

n−1

[

j=0

E(eⁿ_j)

= E(f[eⁿ₀, ..., eⁿ_n−1])

= E(t).

(3) If t=f[t₁, ..., t_n] with t₁, ..., t_n ∈cTτ^lin,(n), E(t_j)∩E(t_k) =∅ for 1≤j < k≤n and assume that E(t_j)⊇E(ˆσ(t_j)) where 1≤j ≤n, then

E(ˆσ[t]) = E(ˆσ[f[t₁, ..., t_n]])

= E(S_nⁿ(σ(f),σ[tˆ ₁], ...,σ[tˆ _n]))

⊆

n

[

j=1

E(ˆσ[tj])

⊆

n

[

j=1

E(t_j)

= E(f[t₁, ..., t_n])

= E(t).

Therefore E(t)⊇E(ˆσ[t]).

Lemma 3.1.6. Let s, t ∈cT_τ and σ ∈ Cohyp(τ). If E(s)∩E(t) = ∅ then E(ˆσ[s])∩ E(ˆσ[t]) =∅.

Proof. Assume that E(s)∩E(t) =∅.

Then ∅ ⊆E(ˆσ[s])∩E(ˆσ[t])⊆E(s)∩E(t) =∅.

Therefore, E(ˆσ[s])∩E(ˆσ[t]) = ∅.

Definition 3.1.7. A linear cohypersubstitution of typeτ is a mapping of cohypersubstitution σ : {f} → cT_τ^lin which preserves the arities. The extension of σ is a mapping ˆ

σ:cT_τ^lin→cT_τ^lin which is inductively defined by the following steps:

(i) σ[eˆ ⁿ_j] :=eⁿ_j for every n ≥1 and 0≤j ≤n−1, (ii) σ[fˆ ] :=σ(f),

16 (iii) σ[fˆ [t₁, ..., t_n]] := Sⁿ_m(σ(f),σ[tˆ ₁], ...,σ[tˆ _n]) and assume that σ[tˆ _j] is already

defined for all 1≤j ≤n.

Note that: By Lemma 3.1.5. and Lemma 3.1.6., we obtain Sⁿ_m(σ(f),σ[tˆ 1], ...,σ[tˆ n]) = S_mⁿ(σ(f),σ[tˆ 1], ...,σ[tˆ n]).

Let Cohyp^lin(τ) be the set of all linear cohypersubstitutions of type τ.

Theorem 3.1.8. The structure(Cohyp^lin(τ);◦coh)is a subsemigroup of(Cohyp(τ);◦coh).

Proof. We will show that σ₁◦_cohσ₂ ∈Cohyp^lin(τ) for all σ₁, σ₂ ∈Cohyp^lin(τ).

Let σ₁, σ₂ ∈Cohyp^lin(τ). Then

(σ₁◦_cohσ₂)[f] = (ˆσ₁◦σ₂)[f]

= ˆσ₁[σ₂(f)]

∈cTτ^lin,(n). Hence, σ1◦cohσ2 ∈Cohyp^lin(τ).

Therefore, Cohyp^lin(τ) is a subsemigroup of Cohyp(τ).

Let σid be the cohypersubstitution defined by σid(f) := f.

Since f is a linear coterm, then σ_id is a linear cohypersubstitution.

Lemma 3.1.9. The linear cohypersubstitution σ_id satisfies the equation σˆ_id[t] = t for all t ∈cT_τ^lin.

Proof. We will prove by induction on the complexity of the linear coterm t.

(1) If t =eⁿ_i where n≥1,0≤i≤n−1, then

ˆ

σ_id[t] = ˆσ_id[eⁿ_i] =eⁿ_i =t.

(2) If t =f, then

ˆ

σ_id[t] = ˆσ_id[f] =σ_id(f) =f =t.

17 (3) If t = f[t1, ..., tn] ∈ cTτ^lin,(n), E(tj)∩E(tk) = ∅ for all 1 ≤ j < k ≤ n and

assume that σˆ_id[t_j] =t_j for all 1≤j ≤n. Then ˆ

σid[t] = ˆσid[f[t1, ..., tn]]

= S_nⁿ(σ_id(f),σˆ_id[t₁], ...,σˆ_id[t_n])

= S_nⁿ(f, t₁, ..., t_n)

= f[t₁, ..., t_n]

= t.

Therefore σˆ_id[t] =t for all t∈cTτ^lin,(n).

Theorem 3.1.10. The structure (Cohyp^lin(τ);◦coh, σid) is a submonoid of (Cohyp(τ);◦_coh, σ_id).

3.2 Idempotent and some regular elements of linear cohypersubstitutions of type τ = (n)

In 2016, D. Boonchari and K. Saengsura were studied the monoid of cohypersubstitutions of type τ = (n) (see [2]). In this section, we determine idempotent and some regular elements of monoid of linear cohypersubstitutions of type τ = (n). Let S be a semigroup, an element a of S is called idempotent if aa = a, and called regular if there exists x∈S such that axa=a (see [4]). For any σ ∈Cohyp^lin(τ)and τ = (n), if σ(f) = t, we denote σ by σ_t. We call the symbol eⁿ_j the injection symbol, for all 0≤j ≤n−1.

Lemma 3.2.1. Let t ∈ cTτ^lin,(n), s₁, ..., s_n ∈ cTτ^lin,(m) and E(s_k) ∩E(s_l) = ∅ for k, l ∈ {1, ..., n} and k 6= l. If E(t) = {eⁿ_j−1|∀j ∈ J,∅ 6= J, J ⊆ {1, ..., n}} and s_j =eⁿ_j−1 for all j ∈J, then S_mⁿ(t, s₁, ..., s_n) = t.

Proof. Assume that E(t) ={eⁿ_j−1|∀j ∈J,∅ 6=J, J ⊆ {1, ..., n}} and s_j =eⁿ_j−1 for all j ∈J.

We give a proof by induction on the complexity of the linear coterm t.

If t =eⁿ_j−1 for some j ∈J, then

Sⁿ_m(t, s₁, ..., s_n) = S_mⁿ(eⁿ_j−1, s₁, ..., s_n)

18

= s_j

= eⁿ_j−1

= t.

If t =f, then

Sⁿ_m(t, s₁, ..., s_n) = S_mⁿ(t, s₁, ..., s_n)

= S_mⁿ(f, s₁, ..., s_n)

= S_mⁿ(f[eⁿ₀, ..., eⁿ_n−1], s₁, ..., s_n)

= f[S_mⁿ(eⁿ₀, s₁, ..., s_n), ..., S_mⁿ(eⁿ_n−1, s₁, ..., s_n)]

= f[s₁, ..., s_n]

= t.

If t =f[t₁, ..., t_n]∈cTτ^lin,(n) and assume that S_mⁿ(t_i, s₁, ..., s_n) =t_i for all i= 1, ..., n.

Then

Sⁿ_m(t, s₁, ..., s_n) = S_mⁿ(t, s₁, ..., s_n)

= S_mⁿ(f[t₁, ..., t_n], s₁, ..., s_n)

= f[S_mⁿ(t1, s1, ..., sn), ..., S_mⁿ(tn, s1, ..., sn)]

= f[t₁, ..., t_n]

= t.

Therefore, S_mⁿ(t, s₁, ..., s_n) =t.

Theorem 3.2.2. Let σ_t ∈Cohyp^lin(n) and t ∈ cTτ^lin,(n). Then σ_t is an idempotent if and only if σˆ_t[t] =t.

Proof. Assume that σ_t is an idempotent. We obtain

ˆ

σ_t[t] = ˆσ_t[σ_t(f)] = (σ_t◦_cohσ_t)(f) = σ_t(f) = t.

Conversely, assume that σˆ_t[t] =t. Then

(σ_t◦_cohσ_t)(f) = ˆσ_t[σ_t(f)]

19

= ˆσ_t[t]

= t

= σ_t(f).

Therefore, σt is an idempotent.

The next result is a condition for an element of Cohyp^lin(n) to be idempotent.

Theorem 3.2.3. Let σ_t ∈ Cohyp^lin(n). If t = eⁿ_i for all 0 ≤ i ≤ n − 1 or t = f[eⁿ₀, ..., eⁿ_n−1], then σt is an idempotent.

Proof. If t =eⁿ_i, then

ˆ

σ_t= ˆσ_t[eⁿ_i] =eⁿ_i =t.

By Theorem 3.2.1., we obtain σ_t is an idempotent.

If t =f[eⁿ₀, ..., eⁿ_n−1], then ˆ

σ_t[t] = ˆσ_t[f[eⁿ₀, ..., eⁿ_n−1]]

= S_nⁿ(σ_t(f),σˆ_t[eⁿ₀], ...,σˆ_t[eⁿ_n−1])

= S_nⁿ(f[eⁿ₀, ..., eⁿ_n−1], eⁿ₀, ..., eⁿ_n−1)

= f[S_mⁿ(eⁿ₀, eⁿ₀, ..., eⁿ_n−1), ..., S_mⁿ(eⁿ_n−1, eⁿ₀, ..., eⁿ_n−1)]

= f[eⁿ₀, ..., eⁿ_n−1]

= t,

then, σt is an idempotent.

Remark 3.2.4. σ_t is not idempotent element if t ∈cTτ^lin,(n)\ {eⁿ_i, f|0≤i≤n−1}.

Example 3.2.5. Let σ_t∈Cohyp^lin(3) where t=f[e³₁, e³₂, e³₀]is a ternary linear coterm of type τ. Then

ˆ

σ_t[t] = ˆσ_t[f[e³₁, e³₂, e³₀]]

= S₃³(σt(f),σˆt[e³₁],σˆt[e³₂],σˆt[e³₀])

= S₃³(t, e³₁, e³₂, e³₀)

= S₃³(f[e³₁, e³₂, e³₀], e³₁, e³₂, e³₀)

20

= f[S₃³(e³₁, e³₁, e³₂, e³₀), S₃³(e³₂, e³₁, e³₂, e³₀), S₃³(e³₀, e³₁, e³₂, e³₀)]

= f[e³₂, e³₀, e³₁] 6= t.

Remark. 1 +n is the number of idempotents of linear cohypersubstitution of type τ = (n).

Now, we characterize all regular elements of Cohyp^lin(n).

Theorem 3.2.6. Let σ_t ∈Cohyp^lin(τ) and t∈ cT_τ. Then σ_t is an regular if and only if σˆt[ˆσs[t]] = t for some σs ∈Cohyp^lin(τ).

Proof. Assume that σ_t is an regular.

There exists σ_s ∈Cohyp^lin(τ) such that σ_t(f) = (σ_t◦_cohσ_s◦_cohσ_t)(f).

Thus

ˆ

σ_t[ˆσ_s[t]] = ˆσ_t[ˆσ_s[σ_t(f)]] = (σ_t◦_cohσ_s◦_cohσ_t)(f) = σ_t(f) = t.

Therefore, σˆt[ˆσs[t]] = t.

Conversely, assume that σˆ_t[ˆσ_s[t]] =t for some σ_s ∈Cohyp^lin(n). Then (σt◦cohσs◦cohσt)(f) = ˆσt[ˆσs[σt[f]]

= ˆσ_t[ˆσ_s[t]]

=t

=σ_t(f).

Therefore, σ_t is an regular.

Theorem 3.2.7. Let σ_t ∈ Cohyp^lin(τ). If t = eⁿ_j for all 0 ≤ j ≤ n−1, then σ_t is regular.

Proof. Assume that t =eⁿ_j for all 0≤j ≤n−1. We obtain ˆ

σ_t[ˆσ_s[t]] = ˆσ_t[ˆσ_s[eⁿ_j]] = eⁿ_j =t.

Therefore, σ_t is regular.