Power Semigroups Of Completely Regular Semigroups

The class of completely regular semigroups is one of the mainly research objects of the algebraic theory of semigroups,and power semigroup of a semigroup is an active topic in the theory of semigroups.A class K of semigroups is said to be globally determined if any two members of K with isomorphic globals are themselves isomorphic.Since the problem of global determinism of semigroups was posed by Tamura in 1967,the global determinism of some subclasses of the class of completely regular semigroups have been proved by many authors.However,whether the class of completely regular semigroups is globally determined global determinism is still an open problem so far.The principal goal of this thesis is to prove the class of completely regular semigroup is globally determined.To this end,we first investigate Green's relations on the power semigroup P(S)of a completely regular semigroup S,and give some characterizations of breakable subsemigroups of S.Based on those,we prove that for any completely regular semigroups S =[Y;S?]and S' =[Y';S'?'],if the power semigroups P(S)and P(S')are isomorphic,then the semigroups S and S' are isomorphic.In particular,for any isomorphism ?:P(S)P(S'),we construct a semilattice isomorphism?:Y ?Y' such that for every ?? Y,the restriction of ? on the subsemigroup P(S?)of P(S)is an isomorphism from P(S?)ontoP(S'?(?)).Secondly,by introducing two operators (?) and (?) between P(S)and P(S'),we study properties of the image of a singleton of P(S)under?.Also,we introduce and study a congruence ?? on the component S? of S,and establish some properties of the image ?(a??)of the ?-class apa under ?.By virtue of those results and the isomorphism ? from P(S)ontoP(S'),we construct an isomorphism ? from S to S'.This shows that the class of completely regular semigroup is globally determined.