Several Questions Of The Half-transitive Semigroups

Let Ln be the symmetric inverse semigroup on the finite set Xn = {1,2,…,n}.In this paper,the composition acts from the left to the right:(t)??=((t)?)?for all ?,? ? In and t ? Xn.Let A be a non-empty proper subset of Xn,and let P= {? ? In |dom(?)(?)Xn\A},Q = {? ? In |dom(?)? A?? and for all x G dom(?)? A,(x)? ? A}.Put M(A)= P ? Q.Clearly,M(A)is a subsemigroup of In.In this paper we obtain firstly that every maximal half-transitive subsemigroup of In is of the form M(A)for A(?)Xn and A ??.Next,we investigate the Greern-relations and Green*-relations on the semigroup M(A),and obtain that M(A)is the type A semigroup.Moreover,we characterize the maximum inverse subsemigroup of M(A,and the inverse sermigroup congruences on M(A)This paper is divided into six parts as followsIn part 1,we introduce briefly the background of development of transfor-mation semigroups,and present the research situations of semi-transitivities and half-transitivities of semigroupsIn part 2,we introduce some basic definitions of semigroups,and important conclusions about the theory of semigroupsIn part 3,we give the definition of maximal half-transitive subsemigroups of In,and obtain the structure,cardinal and number of maximal half-transitive subsemigroups of In.In part 4,we imvestigate Greern-relations and Green*-relations in the half-transitive semigroup M(A).Based on that,we show that M(A)is the type A semigroupIn part 5,we describe the maximum inverse subsemigroup of M(A),and discuss its inverse semigroup congruences on M(A).In part 6,we pose some further related research problems.