The Relationship between Some Regular Subsemigroups of

Abstract

The concept of regular subsemigroups plays an important role in the theory of semigroup. In this work, we study the relationship between some regular subsemigroups on the monoid of all generalized hypersubstitutions of type .

1. Introduction

Let be a semigroup. The class of regular semigroups is one of the most important classes of semigroups. Recall that an element in a semigroup is said to be regular if there exists such that . A semigroup is said to be regular if its element is regular. An element is called idempotent if . It is well-known that an idempotent element is an obvious example of a regular element. We denote the set of all idempotent elements of a semigroup by . Next, we recall some definitions of special elements of a semigroup .

Definition 1. An element of a semigroup is called left (right) regular if there exists such that and is called intraregular if .

Definition 2. An element of a semigroup is called coregular if there is an element such that its coinverse. A semigroup is said to be coregular if each element of is coregular.

Then we have the following proposition.

Proposition 3. Let be a semigroup and . Then is coregular element if and only if .

Proof. () Let be a coregular element in . Then there is an element such that . Thus .
() Suppose that . Then is coregular element.

Definition 4. An element of a semigroup is called antiregular if there exists such that and . The elements and are then called anti-inverse.

Proposition 5. Let be a semigroup and . If is antiregular element, then .

Proof. Let be an antiregular element in . Then there is an element such that and . Thus .

Definition 6. An element of a semigroup is called completely regular if there exists such that and .

Proposition 7 (see [1]). Let be a semigroup and . Then is completely regular element if and only if is both left regular and right regular.

Remark 8. In general, for any semigroup and , we have the following relationship:
is coregular is antiregular is completely regular is left regular, right regular, and intraregular.

The consequent question is as follows: is there a semigroup such that all completely regular elements, left regular elements, right regular elements, or intraregular elements are coregular elements?

As an example, consider the semigroup of all integers with the usual addition. We have that for any nonzero integers and , if and , then but . Then is completely regular but is not coregular.

In this paper, we are interested in the semigroup of generalized hypersubstitutions which is a generalization of hypersubstitutions.

Let be a countably infinite set of symbols called variables and let be an -element set. Let be an indexed set which is disjoint from . Each is called an -ary operation symbol, where is a natural number. Let be a function which assigns every to the number as its arity, written as and is called a type.

An -ary term of type is defined inductively as follows.(i)The variables are -ary terms of type .(ii)If are -ary terms of type , then is an -ary term of type .

The smallest set which contains and is closed under finite application of (ii), denoted by . Let and is called the set of all terms of type .

The concept of a hypersubstitution was introduced first by Denecke et al. in 1991; see [2]. A hypersubstitution of type is a mapping which assigns every -ary operation symbol to an -ary term. The set of all hypersubstitutions of type is denoted by . Every induces a mapping by the following steps:(i), for any variable ,(ii), where are already defined.

A binary operation on is defined by for every where is the usual composition of mappings. Let be the hypersubstitution where . Then is a monoid with as an identity element.

In 2000, Leeratanavalee and Denecke generalized the concepts of a hypersubstitution, a hyperidentity, and a solid variety to a generalized hypersubstitution, a strong hyperidentity, and a strongly solid variety, respectively; see [3]. A generalized hypersubstitution of type is a mapping from the set of all -ary operation symbols into the set of all terms built up by elements of the alphabet and operation symbols from which does not necessarily preserve the arity.

We denote the set of all generalized hypersubstitutions of type by . To define a binary operation on , we defined first the concept of a generalized superposition of terms by the following steps:

for any term ,(i)if , , then ,(ii)if , , then ,(iii)if , then

Every generalized hypersubstitution can be extended to a mapping by the following steps:(i),(ii), for any -ary operation symbol where , are already defined.

We define a binary operation on by where denotes the usual composition of mappings and . Let be the hypersubstitution mapping which maps each -ary operation symbol to the term . Then is a monoid and the monoid of all arity preserving hypersubstitutions of type forms a submonoid of .

Throughout this present paper, we fix the type . By , we denote the generalized hypersubstitution which maps to the term .

Proposition 9 (see [3]). For arbitrary and for arbitrary generalized hypersubstitutions one has (i),(ii).

2. Main Results

Let be a fixed type with the binary operation symbol . Let , the set of all variables occurring in is denoted by  ,  ,  ,  ,  , .

In [4], Puninagool and Leeratanavalee showed that is a set of all regular elements in and .

In [5], Sudsanit et al. proved that the set of all left regular elements and the set of all right regular elements in are the same; that is, .

Furthermore, Boonmee and Leeratanavalee showed that the set of all intraregular elements in is also ; see [6].

Proposition 10. Let . Then the following conditions are equivalent: (a) is coregular,(b) is antiregular,(c) is completely regular,(d) is left regular,(e) is right regular,(f) is intraregular.

Proof. (a) (b) (c) (d) are obtained by Remark 8.
(d) (e) (f) is a consequence of the fact that the set of all left regular, right regular, and intraregular elements in are the same.
(f) (a) Let be intraregular; then . So and then is a coregular element.

Corollary 11. is the set of all coregular elements, antiregular elements, completely regular elements, left regular elements, right regular elements, and intraregular elements in .

Moreover, the next result describes the relationship between some regular subsemigroups of this semigroup.

Let be a semigroup. A nonempty subset of is called a subsemigroup of if . A nonempty subsemigroup of is called a regular subsemigroup if, for any element of , there exists an element such that .

Proposition 12. Let be a regular subsemigroup of . Then the following conditions are equivalent: (a) is coregular,(b) is antiregular,(c) is completely regular,(d) is left regular,(e) is right regular,(f) is intraregular.

Proof. (a) (b) Let be such that . Then is an antiregular element.
(b) (c) Let be an antiregular element in . By using Proposition 5, we have and . Since for all , so is completely regular.
(c) (d) It is obviously by definition.
(d) (e) Since for all , so is right regular.
(e) (f) Let . Since is right regular, so there exists such that . Since , so is intraregular.
(f) (a) Let . Then . So is coregular.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This research was supported by the Human Resource Development in Science Project (Science Achievement Scholarship of Thailand, SAST). The corresponding author is (partially) supported by Faculty of Science, Chiang Mai University, Chiang Mai, Thailand.