A Subdirect Product and Several Sufficient Conditions about a Strong Semilattice of Inverse Semigroups

In this paper, we demonstrate a useful result about the subdirect product of inverse semigroups, firstly. Then we try to add an identity element to each inverse semigroup, in this way, it can be converted into an inverse monoid, at the same time, a semilattice of inverse semigroups can be transformed into a semilattice of inverse monoids. Furthermore, several sufficient conditions about a strong semilattice of inverse monoids are obtained in the end.