finite coverings of semigroups and related structures

For a semigroup s, the covering number of s with respect to semigroups, σs(s), is the minimum number of proper subsemigroups of s whose union is s. this article investigates covering numbers of semigroups and analogously defined covering numbers of inverse semigroups and monoids. our three main theorems give a complete description of the covering number of finite semigroups, finite inverse semigroups, and monoids (modulo groups and infinite semigroups). for a finite semigroup that is neither monogenic nor a group, its covering number is two. for all n ≥ 2, there exists an inverse semigroup with covering number n, similar to the case of loops. finally, a monoid that is neither a group nor a semigroup with an identity adjoined has covering number two as well.