Prime objects in the category of finite semigroups

Determine whether the category of finite semigroups, with primeness understood in Tarski’s sense (Definition 2.1(ii) for nonempty semigroups), contains any prime objects.

Background

The paper proves that the category Semigp of all nonempty semigroups has no prime objects in Tarski’s sense, and extends this non-primeness to several subcategories (commutative semigroups, semigroups with zero, and commutative semigroups with zero).

For finite semigroups, the authors show that Null(n) are not prime using only finite constructions, but their argument that arbitrary non-null semigroups are not prime relies on infinite cardinals. Hence, whether any prime objects exist among finite semigroups remains unresolved.