一类保等价关系部分变换半群的Green关系和正则性

设X为任意集合且X≥3,PX为集合X上的部分变换半群,对于X上的非平凡等价关系E,令PE(X)={f∈PX:(a,b)∈E,(f(a),f(b))∈E},那么PE(X)是PX的一个子半群.从较特殊的情况出发,考虑E为X上的单等价关系,即E=(A×A)∪Δ(X)其中A是X的真子集且A>1,Δ(X)=(x,x):x∈X.给出了PE(X)的正则元的充分必要条件及PE(X)的正则性,刻划了PE(X)的Green关系及PE(X)的正则元之间的Green关系.

Regularity and Green's relations for a class of semigroups of partial transformations preserving an equivalence

Let x denote a set X≥3,PX is a the partial transformation semigroup on the set X,for a non-trivial equivalence E on X,let PE(X)={f∈PX:(a,b)∈E,(f(a),f(b))∈E},then PE(X) is a subsemigroup of PX.This paper can start from a special condition: the single equivalence E on X,namely,E=(A×A)∪Δ(X).A is a proper subset of X,with A>1,Δ(X)=(x,x):x∈X.This paper discusses the sufficient and essential conditions of the regular element and the regularity of PE(X),and describes the Green's relations for PE(X) and the Green's relations between regular elements for PE(X).