On semigroups that are prime in the sense of Tarski, and groups prime in the senses of Tarski and of Rhodes

If C is a category of algebras closed under finite direct products, and MC the commutative monoid of isomorphism classes of members of C, with operation induced by direct product, A. Tarski defined a nonidentity element p of MC to be prime if, whenever it divides a product of two elements, it divides one of them, and defined an object of C to be prime if its isomorphism class has this property. McKenzie, McNulty and Taylor [Citation9, p. 263] ask whether the category of nonempty semigroups has any objects that are prime in this sense. We show in Section 2 that it does not. However, for the category of monoids, and some other subcategories of semigroups, we obtain examples of prime objects in Sections 3–4. In Section 5 two related questions from [Citation9], open so far as I know, are recalled. In Section 6, which can be read independently of the rest of this note, we recall two conditions called primeness by semigroup theorists, and obtain results and examples on the relationships among those conditions and Tarski’s in categories of groups. Section 7 notes a characterization of one of those conditions on finite algebras in an arbitrary variety. Several questions are raised.