保持双向等价关系的变换半群的一些结果

Some Results of Transformation Semigroups that Preserve Double Direction Equivalences

设X为非空集合,E是X上的等价关系,T(X)是X上的全变换半群.记T_(E*)(X)={α∈T(X)|(∀x,y∈X)(x,y)∈E⇔(xα,yα)∈E},则T_(E*)(X)是T(X)的子半群.许多学者对半群T_(E*)(X)作了研究,而沿着这些方向继续研究还可以刻画出T_(E*)(X)中的关系L˜,给出L*=L˜,R=R*的充要条件,证明T_(E*)(X)中的正则元集Re g(T_(E*)(X))形成子半群,同时可以得到Re g(T_(E*)(X))构成完全正则半群和纯正半群的充要条件.

Let X be a non-empty set,E an equivalence relation on X and T(X)the full transformation semigroup on X.Let T_(E*)(X)={α∈T(X)|(∀x,y∈X)(x,y)∈E⇔(xα,yα)∈E},then T_(E*)(X)is a subsemigroup of T(X),which has been studied by many scholars.Continuous study along these directions can describe the L˜-relation of T_(E*)(X),give the necessary and sufficient conditions for L*=L˜and R*=R,and prove that the set of regular elements Reg(T_(E*)(X))of T_(E*)(X)forms a regular subsemigroup.The necessary and sufficient conditions can also be obtained for that Reg(T_(E*)(X))is a completely regular semigroup or an orthodox semigroup.