The Generating Sets And Related Properties Of T(X,?,R)

In the study of transformation semigroups,people are interested in the problem of preserving equivalence relation.For non-empty set X,an equivalence relation p on X,and a cross-sectional R of the partition X/? induced by p.We define T(X,?)= {f ?T(X):(?)V(a,b)??,(af,bf)??},T(X,?,R)= {f?T(X,?):Rf(?)R}.T(X,?)is the semigroup of preserving equivalence relation p.It is clear that T(X,?,R)is a subsemigroup of T(X,?).In recent years,there are many scholars being interested in T(X,?).At the same time,some scholars study T(X,?,R),and get some research results(see[9,10,11]).In 2015,J.Araujo puted forward four questions about the rank of T(X,?,R)in[8].The question 1 is to find the rank of T(X,?,R),when p is a partition in which all of its parts have the same size and T(X,?,R)is regular.The question 2 is to find the rank of T(X,?,R),when p is a partition in which all of its parts have the same size.In this paper,we mainly research problem 1 and 2 which J.Araujo puted forward in[8].For question 1,we give the generating sets of T(X,?,R)when T(X,?,R)is regular.Then the rank of T(X,?,R)is no more than 4.For question 2,we give the generating sets of T(X,?,R).Then the rank of T(X,?,R)is no more than 7.In a sense,we solved the problem 1 and 2.In addition,we consider the maximal regular subsemigroup of T(X,?,R),describe the idempotent elements and the relations L*and R*of T(X,?,R).