Quasi-right semigroups and their properties

A semigroup S is called quasiregular if for every element a in S there exists a natural n and an element x in S such as a~n=a~n×a~n.A semigroup with left central idempotents is a semigroup that may satisfiy x∈y=e×y for any x,y in S and any idempotent e in S.Regular semigroups and abundant semigroups both having left central idempotents have been investigated by Shum and Ren in 1999.Quasiregular semigroups with left central idempotents are studied in this paper. Definition is given to a quasi-right semigroup that a quasiregular semigroup with left central idempotents where the set of its all regular elements is an ideal of S. Such a semigroup is a semilattice of quasi-right groups. Also, some special features of quasi-right semigroups are given.