Several Studies On Regular Subsets And GV-Semigroups

Left regular semigroups,regular subsets and GV-semigroups are studied in this paper.In ChapterⅡ,several equivalent conditions and simple nature of left regular semigroups are given.And a semigroup is a(completely)regular semigroup is equivalent to a(completely)π-regular semigroup under the condition that it is a left regular semigroup.Then the results are promoted to leftπ-regular semigroups.In ChapterⅢ,several relationships between special subsemigroups(eSe,eS, Se) of a semigroup S which has a idempotent are studied.Let T be a subsemigroup of S,and through the definition such as reg(T) = T∪Reg(S),several equations and equivalent conditions such as Reg(eSe) = reg(eSe) = Reg(eS)∩Reg(Se) are proved,some equivalent conditions of Reg(S) = Gr(S) are showen too.Finally, several equivalent conditions are found when the special subsemigroup of S such as ME =∪_e∈E(S)eSe is an(left,right,quasi,bi-side)idea.In ChapterⅣ,the main clues is the local monoid eSe of S.The local GV-semigroup is definded,and some equivalent conditions and nature of GV-semigroups are promoted to local GV-semigroups by using the results of ChapterⅢ.