Abstract
Abstract In this paper, we develop the elementary theory of inverse semigroups to the cases of type B semigroups. The main aim of this paper is to study proper type B semigroups. We introduce first the concept of a left admissible triple. After obtaining some basic properties of left admissible triple, we give the definition of a Q-semigroup and get a structure theorem of Q-semigroup. In particular, we introduce the notion of an admissible triple and give some characterization of proper type B semigroups. It is proved that an arbitrary Q-semigroup with an admissible triple is an E-unitary type B semigroup.
Highlights
Let S be a semigroup and denote the set of idempotents of S by E(S)
We define a ⁎b for the elements a, b of a semigroup S if and only if a, b are related by Green’s relation in T containing S as a subsemigroup. ⁎ is defined dually
Fountain and others investigated some classes of abundant semigroups and got many interesting results
Summary
Let S be a semigroup and denote the set of idempotents of S by E(S). As in [1], the relations ⁎ and ⁎ are generalizations of Green’s relations and , respectively. From [2], a rpp semigroup S is said to be right adequate if E(S) is a semilattice (i.e., any two elements of E(S) commute). A left adequate semigroup is defined dually. A left adequate semigroup S is left type B, if it satisfies the following conditions (LB1) and (LB2): (LB1) for all e, f ∈ E(S1), a ∈ S, (aef )+ = (ae)+(af )+; (LB2) if for all a ∈ S, e ∈ E(S), e ≤ a+, there is an element f ∈ E(S1) such that e = (af )+. A proper inverse semigroup is E-unitary, but the converse is not true. It is an interesting thing to characterize a proper generalization inverse semigroup. We prove that an arbitrary Q-semigroup with an admissible triple is an E-unitary type B semigroup
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
