半群PT_(n)(A;θ)的正则性及格林关系

Regularity and Green's Relations on A Semigroup of PT_(n)(A;θ)

设集合X_(n)={1,2,…,n}并赋予自然序,PT_(n)是集合X_(n)上所有部分变换构成的半群.设A■X_(n)非空,令PT_(n)(A)={α∈PT_(n):imα■A}.在半群PT_(n)(A)上规定运算■:f■g=fθg,则在运算■下,PT_(n)(A)构成一个新的半群,称为它的变种半群.利用正则元及格林关系的定义,讨论了半群PT_(n)(A)的变种半群的正则性,给出了它的变种半群是正则半群的充要条件,并刻画了其变种半群中任意元素间的格林关系,所得结果是半群PT_(n)(A)上相应结果的推广.

Let set X_(n)={1,2,…,n}with the natural order.PT_(n) is a semigroup consisting of all partial transformations on set X_(n).Let A■X_(n) non-empty and PT_(n)(A)={α∈PT_(n):imα■A}.Obviously PT_(n)(A)is a subsemigroup of PT_(n).Specify the operation■on the semigroup PT_(n)(A)then under the operation■,PT_(n)(A)forms a semigroup,called its variant semigroup.Using the definitions of regular elements and Green's relations,the regularity of the variant semigroup of semigroup PT_(n)(A)is discussed.The sufficient and necessary conditions for its variant semigroup to be a regular semigroup are given.The Green's relationship between any elements on the variant semigroup of semigroup PT_(n)(A)is characterized,and the result is a generalization of the corresponding results on the semigroup PT_(n)(A).