On The Properties Of Tow Subsemigroups Of The Transformation Semigroup

Let X be a nonempty set. The set T_X of all transformations on X, acting on the left, is a semigroup with the operation of mappings compassion, called the full transformation semigroup on X. The set I_X consisting of all bijections between subsets of X (together with an empty mapping) is also a semigroup, called the symmetric inverse semigroup, with the operation defined byLet E be an equivalence on X. The E-preserved transformation semigroup on X is thatThe E^*-preserved partial bi-transformation semigroup on X is defined asIn this paper, we consider the special case that X = {1,2,…,n}. Let E be an equivalence on X, denote X|E the set of all E-classes.f∈T_E(X) is called a preserving orientation transformation on E classes, if(?)A∈X|E,write A ={α₁,α₂,…,α_n) withα₁ <α₂ <…<α_n, (f(α₁),f(α₂),…,f(α_n)) is a cyclic sequence, that is, there exists at most one i such that f(α_i)>f(α_i+1).The set OPP_E(X) of all preserving orientation transformation on E classes is a subsemigroup of T_E(X),namely OPP_E(X)={f∈T_E(X):f is preserving orientation transformation on E classes}.In Chapter 2 we mainly consider the Green's relations and regularity of OPP_E(X). The main conclusions are given as follwing:Theorem2.1 give the two equivalent conditions of R's relation and theorem2.1 educe the equivalent conditon of L's relation.Moreover theorem 2.4 give the equivalent condition of D's relation. Theorem 2.6 and Theorem 2.7 consider the sufficient and essential condition of the regular elements and the regularity of OPP_E(X).respectively.In Chapter three we discuss the rank of OPP_E(X).In this Chapter,let X ={1,2,…,mn},where m>2,n> 3.The equivalence on E on X is always defined by where A_i = [(i-1)n+1,in](1≤i≤m).we discuss the rank of OPP_E(X) and the rank of some subsemigroup of OPP_E(X).The main result is given as follwing:Theorem 3.6 OPP_E(X)=<α,β,a,Ï„,r>.a = (1,2,…, n),r = (?) is a idmepotent in T_X which takes 2 to 1,and fixes the other points.In Chaputer four we consider the rank and maximal inverse subsemigroup of I_E^*(X).Theorem4.2 and theorem4.2 give the equivalent condition of Green's relations.Let b = (12) be a permutation on X,which takes 2 to 1,takes 1 to 2,and fixed the other points except for 1,2,V_r = {f∈I_E(X),|imf| = r},0≤r≤nm.Theorem 4.7 Let f be an arbitrary element of V_nm-1,I_E^*(X)=<α,β,a,b,f>.Theorem 4.8 Let S be an maximal inverse subsimgroup of I_E^*(X).Then S is one of following:(a) S =K_nm-2∪V_nm(b) S =K_nm-1∪G,G is a maximal subgroup of V_nm.