Proper covers for LU-ample type B semigroups

Abstract

A LU-semiadequate semigroup is a semigroup whose projections commute and in which each LU- class contains a projection. Let (S,U) be a LU-semiadequate semigroup, whereU is the set of projections of S. It is the fact that each LU-class of (S,U) contains a unique projection. For an element a of (S,U), the projection in the LU-class containing a is denoted by a∗. Let 1 be an identity of S1, U1=U ∪{1}. If (S,U) satisfying: (1) for all e, f in U1and all elements a in (S,U), (efa)∗= (ea)∗(fa)∗; (2) for all elements a in (S,U) and all projections e ≤ a∗, there is an element f in U1such that e = (fa)∗, then we say that (S,U) is a LU-ample type B semigroup. In this paper, we introduce the concept of a proper cover of a LU-ample type B semigroup and prove that any proper cover for a LU-ample type B semigroup is a proper cover over a monoid. A structure theorem of proper covers for LU-ample type B semigroups is obtained. This theorem generalizes the result of Li-Wang for right type B semigroups.