Eventual regularity and isomorphism theorems of some regressive transformation semigroups

Abstract

A partial transformation [alpha] on a poset is regressive if x[alpha][is less than or equal to] x for all x [is equivalent to] dom [alpha]. For a poset X, let P[subscript RE](X),I[subscript RE](X) and T[subscript RE](X) denote respectively the regressive partial transformation semigroup on X, the regressive 1-1 partial transformation semigroup on X and the full regressive transformation semigroup on X. The following results relating to regularity and eventual regularity are known. The semigroup P[subscript RE](X),[I[subscript RE](X)] is regular if and only if X is isolated, and T[subscript RE](X) is regular if and only if [is less than or equal to] 2 for every subchain C of X. If S(X) is P[subscript RE](X),T[subscript RE](X) or I[subscript RE](X), then S(X) is eventually regular if and only if there is a positive integer n such that [is less than or equal to]n for every subchain C of X. A. Umar has proved an important isomorphism theorem as follows :For chains X and Y, T[subscript RE](X)[is equivalent to] T[subscript RE](Y) if and only if X and Y are order-isomorphic. Our purpose is to extend the above known results. The following subsemigroups of P[subscript RE](X),I[subscript RE](X) and T[subscript RE](X) are considered where X' is a subposet of X. P[subscript RE](X,X')={[alpha][is equivalent to]P[subscript RE](X) an[alpha][is less than or equal to]X'}, P[subscript RE](X,X]) = {[alpha][is equivalent to]P[subscript RE](X) '[alpha][is less than or equal to]X'} and I[subscript RE](X,X'),I[subscript RE](X,X'), T[subscript RE](X,X') and T[subscript RE](X,X') are defined similarly. We characterize when the semigroups P[subscript RE](X,X'), I[subscript RE](X,X'), T[subscript RE](X,X'), P[subscript RE](X,X'), I[subscript RE](X,X') and T[subscript RE](X,X') are regular, and then the above known results of regularity become our special cases. For eventual regularity the known result mentioned above and one of its lemmas are used to obtain necessary and sufficient conditions for all of these semigroups to be eventually regular. Our main isomorphism theorems are as follows: If P[subscript RE](X,X')[is equivalent to]P[subscript RE](Y,Y') then X' and Y' are order-isomorphic. If I[subscript RE](X,X')[is equivalent to] I[subscript RE](Y,Y') then X' and Y' are order-isomorphic. In particular, P[subscript RE](X)[is equivalent to]P[subscript RE](Y) if and only if X and Y are order-isomorphic, and also [I[subscript RE](X)][is equivalent to]I[subscript RE](Y] if and only if X and Y are order-isomorphic. If X and Y are chains and T[subscript RE](X,X')[is equivalent to]T[subscript RE](Y,Y'), then X' and Y' are order-isomorphic. It can be seen that the last isomorphism theorem extends Umar's Isomorphism Theorem.