TY - JOUR T1 - A Class of Left $E$-Adequate Semigroups AU - Li , Yonghua AU - He , Yong JO - Communications in Mathematical Research VL - 4 SP - 289 EP - 303 PY - 2021 DA - 2021/05 SN - 26 DO - http://doi.org/ UR - https://global-sci.org/intro/article_detail/cmr/19127.html KW - type $µ^+$ semigroup, abundant semigroup, left $E$-adequate semigroup, $E^+$-product. AB -

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.