Some Studies On Finite Transformation Semigroups Of Weak Y-Stabilizer

Let T(X) be the full transformation semigroup on a finite set X and Y an arbitrary nonempty subset of X. We let Obviously. W (Y) is a subsemigroup of T (X). We call it the finite transformation semigroup of weak Y-stabilizer. In particular, if|Y|=|X|, we have W(Y)=T(X); if 1=|Y|<|X|, we have W(Y)(?)PT(X\Y), where PT(X\Y) is the partial transformation semigroup on X\Y. In this thesis, we mainly investigate some congruence relations on the transformation semigroup W(Y), such as maximal congruence, minimal congruence and maximum idempotent-separating congruence. In addition, for eachα∈W(Y), we also characterize all weak inverses of a in W(Y).This dissertation consists of three chapters in all. In chapter 1. some basis concepts on semigroup algebra theory and all well-known results and conclusions on the transformation semigroup W(Y) are proposed. In chapter 2, after showing all weak inverses of eachα∈W(Y), we obtain the maximum idempotent-separating congruence on W(Y). In chapter 3, we completely describe all maximal congruences and minimal congruences on the transformation semigroup W(Y).