Generalized transformation semigroups and linear transformation semigroups whose bi-ideals are quasi-ideals

Abstract

A subsemigroup Q of a semigroup S is called a quasi-ideal of S if SQ intersection QS Q. By a bi-ideal of S we mean a subsemigroup B of S such that BSB B. Quasi-ideals are a generalization of left ideals and right ideals and bi-ideals generalize quasi-ideals. The notion of bi-ideal and the notion of quasi-ideal for semigroups were introduced respectively By R. A. Good and D.R. Huges in 1952 and O. Steinfeld in 1956. Since then, both quasi-ideals and bi-ideals of semigroups have been widely studied. Semigroups whose bi-ideals and quasi-ideals coincide are of our interest in this research. One calls such semigroups BQ-semigroups. For sets X and Y, let P(X,Y) be the set of all mappings alpha : A -> Y where A X. For theta P(Y, X), let (P(X,Y), theta) denote the semigroup (P(X,Y),*) where alpha*beta = alpha theta beta for all alpha, beta P(X,Y). The first purpose of this research is to characterize when certain subsemigroups of (P(X,Y), theta) with a particular theta are BQ-semigroups in terms of the cardinalities of X and Y. For a vector space V over a division ring, let (L(V) be the semigroup under composition of all linear transformations alpha : V -> V. Various subsemigroups of L(V) defined by kernels and images of linear transformations are studied for our second purpose. We characterize when these linear transformation semigroups are BQ-semigroups in terms of the dimensions of V. Finally, we study the full order-preserving transformation semigroup TOP (I) on an interval I of real numbers. Necessary and sufficient conditions for I so that TOP(I) is a BQ-semigroup are given.