(?)-abundant Semigroups

A cancellaive congruence is defined as a congruence p on semigroup S in which S/p is cancellative.A semigroup S is called E-abundant if S satisfies congruence condition and every (?)-class and every (?)-class of S contains some idempotents.Certainly,the class of E-abundant semigroups is a generalization both the class of abundant semigroups and the class of regular semigroups.In recent years,E-abundant semigroups become a significant area of the study of semigroup theory.In this paper,we majorly research a special class of E-abundant semigroups,namely,(?)-abundant semigroups.An E-abundant semigroup(S,E)will be called (?)-abundant if every minimum cancellative congruence class of(S,E)contains greatest element with respect to the natural partial order relation.In fact,(?)-abundant semigroups not only is a generalization of the F-regular semigroups in the class of regular semigroups but also is an extending of F-abundant semigroups in the class of abundant semigroups.Firstly,the natural partial order relation on an E-semiabundant semigroup is defined,and the minimum cancellative congruence σ is discussed,furthermore,a concretely algebraic expression of σ is given.Next,we introduce (?)-abundant semigroup and strongly (?)-abundant semigroup and obtain some basic properties of them.Finally,the (?)-system of a semigroup is defined in virtue of the r-isomorphism on a set,and further (?)(M,E;Φ,Ψ)as an algebraic structure of strongly (?)-abundant semigroup is established.It is proved the consequence as follow:if(M,E;Φ,Ψ)be a (?)-system,then (?)(M,E;Φ,Ψ)is a strongly (?)-abundant semigroup.Conversely,every strongly (?)-abundant semigroup is isomorphic to one (?)(M,E;Φ,Ψ)constructed in this way.Our work not only generalizes the results of C.C.Edwards for F-regular semigroups but also generalizes the results of X.J.Guo for F-abundant semigroups.