Regressive generalized transformation semigroups

Abstract

For a set X, let P(X), T(X) and I(X) denote respectively the partial transformation semigroup on X, the full transformation semigroup on X and the one-to-one partial transformation semigroup on X. Also, let AP(X) = {[alpha] [is an element of] P(X) [alpha] is almost identical} and define AT(X) and AI(X) similarly. Then AP(X), AT(X) and AI(X) are subsemigroups of P(X), T(X) and I(X), respectively. We generalize a transformation semigroup on X (a subsemigroup of P(X)) to be a semigroup (S(X), [theta]) where S(X) is a transformation semigroup on X, [theta] [is an element of] S[superscript 1](X) and ( S(X), [theta]) = (S(X),*) where [alpha]* [beta] = [alpha] [theta] [beta] for all [alpha], [beta] [is an element of] S(X). For a poset X, let P[subscript RE](X) = {[alpha][is an element of] P(X) | [alpha] is regressive}, and T[subscript RE](X), I[subscript RE](X), AP[subscript RE](X), AT[subscript RE](X) and AI[subscript RE](X) are defined similarly. Then P[subscript RE](X), T[subscript RE](X), I[subscript RE](X), AP[subscript RE](X), AT[subscript RE](X) and AI[subscript RE](X) are respectively subsemigroups of P(X), T(X), I(X), AP(X), AT(X) and AI(X). The following facts are known. If S(X) is P[subscript RE](X), I[subscript RE](X), AP[subscript RE](X) or AI[subscript RE](X), then S(X) is regular if and only if X is isolated. If S(X) is T[subscript RE](X) or AT[subscript RE](X), then S(X) is regular if and only if | [is less than or equal to] 2 for every chain C of X. If S(X) is P[subscript RE](X), T[subscript RE](X) or I[subscript RE](X), 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 chain C of X. Moreover, every regressive almost identical transformation semigroup on X (every subsemigroup of AP[subscript RE](X) ) is eventually regular. The purpose of this research is to generalize all the above known results by considering those on the semigroup (S(X), [theta]) with [theta] S[superscript 1](X) where S(X) is a regressive transformation semigroup on X of our purpose. In addition, some isomorphism theorems on regressive generalized transformation semigroups are provided.