On semigroups that are prime in the sense of Tarski, and groups prime in the senses of Tarski and of Rhodes
George M. Bergman
TL;DR
The paper investigates primeness notions for semigroups and groups in the sense of Tarski and Rhodes, examining when objects can be prime with respect to direct (and semidirect) product decompositions. It proves a striking nonexistence result for prime objects in the category of nonempty semigroups, while establishing positive primeness results for cancellative semigroups and certain monoid settings (notably that $ ext{N}+k$ is prime in appropriate categories). It then contrasts Tarski-primeness with Rhodes-primeness in groups, providing explicit constructions showing their separation and leveraging monolithic structure and Krull–Schmidt theory in the finite case; it also offers a general criterion for Rhodes-primeness in arbitrary varieties of algebras. The work culminates with a variety-wide perspective, connecting Rhodes-primeness to join-primes in subvariety lattices and raising several open questions for further study.
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.