一类夹心变换半群的正则元

设T_X是非空集合X上全变换半群,E是X上等价关系,则T_?(X)={f∈T_X:?_x,y∈X,(f(x),f(y))∈E?(x,y)∈E}是T_X的反射等价关系的子半群.取定θ∈T_?(X),在T_?(X)上定义新的运算°为f°g=fθg,其中fθg表示一般意义上映射f、θ、g的复合.关于这个运算°,T_?(X)成为夹心变换半群T_?(X;θ).本文刻画了它的正则元,给出了T_?(X;θ)是正则半群的充要条件.

Regular Elements of a Kind of Sandwich Transformation Semigroups

Let T_X be the full transformation semigroup on a nonempty setXandEbe an equivalence onX. ThenT_?(X)={f∈TX:?x,y∈X,(f(x),f(y)) ∈E(x,y) ∈E}is a subsemigroup of T_X of transformations reflecting the equivalenceE. Fix an elementθ∈T_?(X) and define an opera-tion ° onT_?(X) byf°g = fθg wherefθgdenotes the composition of the mapsg,θandfin the usual sense. With respect to the new operation °,T_?(X) forms a new semigroup which is called a sandwich semigroup ofT_?(X) and denoted by T_?(X;θ). The regular elements of the sandwich transformation semigroupT_?(X;θ) were characterized and a necessa-ry and sufficient condition for the regular semigroupT_?(X;θ) was presented.