Some Studies On Two Classes Of Transformation Semigroups

In this paper,two kinds of transformation semigroups,namely,the monoid of partial signed permutations and the symmetric inverse semigroup,are studied.Let I =[n]= {1,2,...,n},-I = {-x |x∈I},where n is a positive integer.The monoid of partial signed permutations I±I is a set of all partial signed permutations of the set I under the composition of transformations.We study the principle left ideals,principle right ideals and principle ideals of I±I,then we get the Green relation of I±I further we determine the regularity and completely regularity of I±I.The finite symmetric inverse semigroup In,on a finite set[n]={1,2,…,n｝is the set of all one-to-one partial transformation of[n]with composition of transformations as multiplication.B.M.Schein and B.Teclezghi described all the endomorphisms of the symmetric inverse semigroups in 1997.Surprisingly,nobody considered the homomorphisms of two different symmetric inverse semigroups.We describe all the homomorphisms from the symmetric inverse semigroup I,to the symmetric inverse semigroup Im with n>m≥1 and count their number.