The Structure Of The Semigroup With Some Inner Property

In this paper,we discuss the structures of inner-regular semigroup, inner - orthodox semigroups,and inner - inverse semigroups. The main results are the following theorems: Theorem 1: S is inner -regular semigroup,then S M(2,r),r=| S| -1 . Theorem 2: The structure of closed -regular semigroup S generated by two idempotents e and f is S=, (ef)m=ef, (fe)m =fe, where m = min{ i:(ef)i=ef, i>1, i∈N}.Theorem 3: The structure of inner -orthodox semigroups S must be one of the following two types:(1) If S is non-regular semigroup, then S ≌M(2,r), r=|s|-1.(2) If S is regular semigroup, then S= , e2=e , f2=f,(ef)m-1l= e, f(ef)m-1=f, m > 2 and m ∈ Z .Theoroem 4: Let S be an inner - inverse semigroup.(1) IfS is a non-regular semigroup, then S ≌M (2,r), r =|s|-1.(2) If S is a regular semigroup, then S = , e2=e f2 =f, ef=e, fe=f(or ef=f, fe=e).