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

TL;DR

This paper investigates prime objects in categories of algebras, especially semigroups and monoids, exploring their existence, properties, and relationships with other primeness conditions in algebraic structures.

Contribution

It provides examples of prime objects in monoids and subcategories, answers a question about semigroups, and explores relationships among different primeness notions in groups.

Findings

No prime objects in the category of nonempty semigroups.

Existence of prime objects in monoids and some subcategories.

Relationships among primeness conditions in groups are characterized.

Abstract

If $\mathcal{C}$ is a category of algebras closed under finite direct products, and $M_\mathcal{C}$ the commutative monoid of isomorphism classes of members of $\mathcal{C},$ with operation induced by direct product, A.Tarski defined a nonidentity element $p$ of $M_\mathcal{C}$ to be prime if, whenever it divides a product of two elements in that monoid, it divides one of them, and called an object of $\mathcal{C}$ prime if its isomorphism class has this property. McKenzie, McNulty and Taylor ask whether the category of nonempty semigroups has any prime objects. 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 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 related conditions that are called primeness by semigroup theorists, and obtain results and examples on the relationships among those two conditions and Tarski's, in categories of groups. Section 7 notes an interesting characterization of one of those conditions when applied to finite algebras in an arbitrary variety. Various questions are raised.