A Class of Left $E$-Adequate Semigroups (Communications in Mathematical Research, 2021)
In this paper we establish a construction of a class of left $E$-adequatesemigroups by using semilattices of cancellative monoids and fundamental left $E$-adequate semigroups. We first introduce concepts of type $µ^+$ ($µ^∗$, $µ$) abundantsemigroups and type $µ^+$ left $E$-adequate semigroups. In fact, regular semigroups aretype $µ^+$ abundant semigroups and inverse semigroups are type $µ^+$ left $E$-adequatesemigroups. Next, we construct a special kind of algebras called $E^+$-product. It isproved that every $E^+$-product is a type $µ^+$ left $E$-adequate semigroup, and everytype $µ^+$ left $E$-adequate semigroup is isomorphic to an $E^+$-product of a semilatticeof cancellative monoids with a fundamental left $E$-adequate semigroup. Finally, as acorollary of the main result, it is deduced that every inverse semigroup is isomorphicto an $E^+$-product of a Clifford semigroup by a fundamental inverse semigroup.
Keywords: type $µ^+$ semigroup; abundant semigroup; left $E$-adequate semigroup; $E^+$-product.