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.
