(untitled) (Communications in Mathematical Research, 2021)
In this paper we establish a construction of a class of left E-adequate semigroups by using semilattices of cancellative monoids and fundamental left E-adequate semigroups. We first introduce concepts of type µ^+ (µ^∗, µ) abundant semigroups and type µ^+ left E-adequate semigroups. In fact, regular semigroups are type µ^+ abundant semigroups and inverse semigroups are type µ^+ left E-adequate semigroups. Next, we construct a special kind of algebras called E^+-product. It is proved that every E^+-product is a type µ^+ left E-adequate semigroup, and every type µ^+ left E-adequate semigroup is isomorphic to an E^+-product of a semilattice of cancellative monoids with a fundamental left E-adequate semigroup. Finally, as a corollary of the main result, it is deduced that every inverse semigroup is isomorphic to an E^+-product of a Clifford semigroup by a fundamental inverse semigroup.