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

1. Problem Statement and Context

The paper investigates the concept of primality in algebraic categories, specifically focusing on semigroups and groups. The central definition, attributed to A. Tarski, defines an object $P$ in a category $\mathcal{C}$ (closed under finite direct products) as prime if its isomorphism class $[P]$ in the commutative monoid of isomorphism classes is a prime element. Formally, if $P$ is a direct factor of a product $Y = Y_0 \times Y_1$, then $P$ must be a direct factor of either $Y_0$ or $Y_1$.

The paper addresses three specific questions raised in the literature (specifically in McKenzie, McNulty, and Taylor [9]):

1. Does the category of nonempty semigroups contain any Tarski-prime objects?
2. Are there Tarski-prime objects in the category of finite semigroups?
3. How does Tarski-primeness relate to two other notions of primeness used by semigroup theorists (Rhodes-primeness with respect to direct and semidirect products) when applied to groups?

2. Methodology

Bergman employs a combination of structural algebraic analysis, construction of counterexamples, and the use of specific algebraic tools:

• Action Equivalence and Null Semigroups: To disprove primeness in general semigroups, the author utilizes the concept of action equivalence (elements $x, x'$ are equivalent if $xy=x'y$ and $yx=yx'$ for all $y$) and null semigroups (where all products are equal).
• Skeletons and Isomorphism Lemmas: The paper introduces "skeletons" (minimal subsemigroups containing representatives of action equivalence classes) to establish isomorphisms between products involving null semigroups (e.g., $S \times \text{Null}(\kappa) \cong S^+ \times \text{Null}(\kappa)$).
• Extension of Homomorphisms: For positive results in restricted categories, the author uses Lemma 3.1, which allows extending homomorphisms from an ideal to the whole semigroup when the codomain is a group. This is crucial for analyzing cancellative semigroups.
• Group Theory Tools: For the group-theoretic sections, the paper utilizes the Krull-Schmidt Theorem, properties of monolithic groups (groups with a unique minimal normal subgroup), and the theory of wreath products (Kaloujnine-Krasner Theorem).

3. Key Contributions and Results

A. Non-Existence of Prime Objects in General Semigroups (Sections 2 & 5)

• Main Result: The category of nonempty semigroups (Semigp) contains no Tarski-prime objects.
• Proof Strategy: The author constructs counterexamples for any candidate semigroup $S$.
  • If $S$ is non-null, the author constructs $S^+$ (adding elements to an action equivalence class) and shows $S \times \text{Null}(\kappa) \cong S^+ \times \text{Null}(\kappa)$. Since $S$ is not a factor of $S^+$ (due to cardinality differences in action classes) nor of $\text{Null}(\kappa)$, $S$ is not prime.
  • If $S$ is a null semigroup $\text{Null}(\kappa)$, the author constructs a specific semigroup $S$ such that $S \times \text{Null}(n)$ has $\text{Null}(\kappa)$ as a factor, but neither $S$ nor $\text{Null}(n)$ does.
• Corollaries: This result extends to the categories of commutative semigroups and semigroups with a zero element.
• Open Question: The paper leaves open whether the category of finite semigroups contains prime objects, as the proofs for the general case rely on infinite cardinals.

B. Positive Results in Restricted Categories (Sections 3 & 4)

• Cancellative Semigroups: The author proves that for any non-negative integer $k$, the additive semigroup $\mathbb{N} + k$ (integers $\ge k$) is Tarski-prime in the categories of:
  • Cancellative semigroups.
  • Right/Left cancellative semigroups.
  • Weakly cancellative semigroups (where no two distinct elements are action equivalent).
• Monoids: The semigroup $\mathbb{N}$ is prime in the category of monoids and in the category of semigroups where every element is a product.
• General Monoids: The paper references a known result (Exercise 5.15 in [9]) stating that in varieties of algebras with a zero and a binary operation, any nontrivial object with a finite center that is not a direct product is prime. This implies many submonoids of $\mathbb{N}$ and free monoids are prime.

C. Comparison of Primeness Notions in Groups (Section 6)

The paper distinguishes between three definitions of primeness for groups $G$:

1. Tarski-prime: $G$ is a direct factor of $H \implies G$ is a factor of one of the components of $H$.
2. Rhodes-prime (Direct): $G$ is a subquotient of $H_0 \times H_1 \implies G$ is a subquotient of $H_0$ or $H_1$.
3. Rhodes-prime (Semidirect): Same as above but for semidirect products $H_0 \rtimes H_1$.

Key Findings for Groups:

• $\mathbb{Z}$ is not Tarski-prime: The additive group of integers is not Tarski-prime in the category of groups. A counterexample is constructed using $(\mathbb{Z}_p \rtimes \mathbb{Z}) \times (\mathbb{Z}_q \rtimes \mathbb{Z})$, which can be rearranged to show $\mathbb{Z}$ is a direct factor, even though it is not a factor of the original components.
• $\mathbb{Z}$ is Rhodes-prime: $\mathbb{Z}$ is Rhodes-prime with respect to both direct and semidirect products because it is a subquotient of a group iff that group has an element of infinite order.
• Finite Groups:
  • Tarski-prime: By Krull-Schmidt, any finite group that is not a nontrivial direct product is Tarski-prime.
  • Rhodes-prime (Semidirect): A finite group is Rhodes-prime with respect to semidirect products if and only if it is simple (Proposition 6.5). This contrasts with the infinite case where $\mathbb{Z}$ is Rhodes-prime but not simple.
  • Rhodes-prime (Direct): A finite group is Rhodes-prime with respect to direct products if it is monolithic (has a unique minimal normal subgroup) and satisfies additional conditions (e.g., the monolith is non-abelian, or the group is a semidirect product of the monolith).
• Implications: The paper establishes a hierarchy for finite groups:
  $$\text{Rhodes-semidirect} \implies \text{Rhodes-direct} \implies \text{Monolithic} \implies \text{Tarski-prime}$$
  None of these implications are reversible, as demonstrated by specific counterexamples (e.g., $S_n$ is Rhodes-direct but not Rhodes-semidirect; certain $p$-groups are monolithic but not Rhodes-direct).

D. General Varieties (Section 7)

The paper provides a characterization of Rhodes-primeness for finite algebras in arbitrary varieties. An object $X$ is Rhodes-prime with respect to direct products if and only if the subvariety generated by $X$ is join-prime in the lattice of subvarieties generated by finite algebras, and $X$ is a subquotient of any algebra generating a subvariety containing it.

4. Significance

• Resolution of Open Problems: The paper definitively answers the question of whether prime semigroups exist in the general category (No) and provides a rich class of examples where they do exist (cancellative semigroups, monoids).
• Clarification of Concepts: It rigorously distinguishes between Tarski-primeness (factorization of direct products) and Rhodes-primeness (factorization of subquotients), showing that they behave very differently in the context of groups (e.g., $\mathbb{Z}$ is Rhodes-prime but not Tarski-prime).
• Structural Insights: The results on finite groups link primeness to deep structural properties like simplicity and monolithicity, offering a new perspective on the decomposition of finite groups.
• Open Questions: The paper raises several open questions, most notably whether the category of finite semigroups contains prime objects, and the reversibility of implications between different primeness notions in infinite groups and general varieties.

In summary, Bergman's work provides a comprehensive map of primeness in algebraic structures, proving the non-existence of primes in general semigroups while identifying robust classes of primes in cancellative settings, and establishing a nuanced hierarchy of primeness conditions for finite groups.