The Relationship between Some Regular Subsemigroups of

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