| A semigroup S is said to be a r-wide semigroup if S is ￡*,～-abundant and R*,～-abundant,that is to say,every￡*,～-class and R*,～-class of S contain the element of E.In recent years,as a class of generalized regular semigroups,r-wide semigroups have been widely concerned and studied by many people.This paper aims to study some special r-wide semigroups,namely￡*,～-inverse semigroup.A r-wide semigroup S having the congruence condition is said to be an￡*,～-inverse semigroup if S is a TC semigroup whose set of idempotent E forms a left regular band,i.e.fef=fe for any e,feS.The￡*,～-inverse semigroup is the common generalization of the left inverse semigroup and ￡*-inverse semigroup.This paper first discuss the basic properties of r-wide semigroup and the(*,～)-good congurence and(￡*,～)-good morphism on this semigroup.And then we give the definition of TC condition,next deduce that the following equivalent characterization of r-wide semigroup which satisfying the TC condition:let S be a r-wide semigroup whose set of idempotent E forms a left regular band.Then S satisfying TC condition if and only if for any element a in S,the following two conditions hold:i)for any a+ and h∈ω(a+),there existsg∈<a*)such that ha=ag;ii)for any a*and e∈ω(a*),there exists f∈<a+>such that ae=fa.Secondly,this paper give the definition of the￡*,～-inverse semigroup,researched the related properties of the￡*,～-inverse semigroup and the(*,～)-good congruence on the ￡*,～-inverse semigroup.At the same time,this paper defined the concept of the left tache product of semigroup and then researched the related properties of the left tache product of semigroup.Finally,a structure theorem of the ￡*,～-inverse semigroup is established,which proved that every left tache product of a left regular band and a type T semigroup is a￡*,～-inverse semigroup.Conversely,a￡*,～-inverse semigroup is isomorphic to a left tache product of a left regular band and a type T semigroup.Moreover,we give an example for the ￡*,～-inverse semigroup by using the left tache product. |
