Some local Subsemigroups of semigroups of partial transformations and linear transformations

Abstract

The set of all idempotents of a semigroup S is denoted by E(S) . A local subset of a semigroup S is a subset of S of the form eAe where e E(S) and A is a subsemigroup of S . A local subset of S need not be a subsemigroup of S . By a local subsemigroup of S we mean a local subset of S which is a subsemigroup of S . Notice that for every e E(S) , the local subset eSe is a local subsemigroup of S . A semigroup S is regular if for every a S , a = axa for some x S. Let be a nonempty set. Denote by P(X) , T(X) , I(X) and G(X) the partial transformation semigroup, the full transformation semigroup, the 1-1 partial transformation semigroup (the symmetric inverse semigroup) and the symmetric group on X, res-pectively. In this research, it is shown that for any a E(P(X)) , aT(X)a is a local subsemigroup of P(X) and we characterize a E(P(X)) when X is finite for which the local subsets aI(X)a and aG(X)a of P(X) are local subsemigroups of P(X) . These characterizations automatically imply that these local subsemigroups of P(X) are regular semigroups. We also study the semigroup L(V) of all linear transformations of a finite-dimensional vector space V in the same maner. We provide a necessary and sufficient condition for a e(L(V)) guaranteeing that aGL(V)a is a local subsemigroup of L(V) where GL(V) is the group of all isomorphisms of V . In addition, the local subset AGn(F) of the full nxn matrix semigroup Mn(F) over a field is considered similarly where Gn(F) is the group of all nonsingular nxn matrices over F . These local subsemigroups of L(V) and Mn(F) are also regular.