Maximal Pseudovarieties of Finite Semigroups

• Maximal pseudovarieties of finite semigroups are classes defined by closure under finite direct products, subsemigroups, and homomorphic images, characterized by expansion or permutation identities.
• Expansion-identity pseudovarieties enforce almost complete regularity through idempotent power operations, while permutation-identity pseudovarieties allow invariance under fixed nontrivial permutations.
• The inclusion of the nilpotent semigroup T serves as a key obstruction, determining whether a uniform permutation identity can be globally satisfied within the lattice of pseudovarieties.

A maximal pseudovariety of finite semigroups is a class of finite semigroups defined by closure properties under finite direct products, subsemigroups, and homomorphic images, and which—within a given equational or combinatorial context—is not properly contained in any larger pseudovariety without violating a key structural or identity-based constraint. Recent research, notably (Thumm, 23 Sep 2025), presents a comprehensive classification of such maximal pseudovarieties arising from product identities, providing insight into their structure, the sharp dichotomy they induce, and their role in the broader lattice of finite semigroup pseudovarieties.

1. Product Identities and the Maximal Pseudovariety Dichotomy

The central setting is identities of the form

$x_1 x_2 \cdots x_n \approx \rho(x_1, \ldots, x_n)$

where $\rho$ is an $n$-ary term built from $x_1, \ldots, x_n$. The main structural result asserts that any nontrivial product identity of this form forces the finite semigroups (or pseudovarieties) that satisfy it into two qualitatively distinct maximal classes:

1. Expansion Identity Type: There exist $1 \leq i \leq j \leq n$ such that the identity

$$x_1 x_2 \cdots x_n \approx x_1 \cdots x_{i-1} (x_i\cdots x_j)^{\omega+1} x_{j+1} \cdots x_n$$

holds in all members. Here, “$(\omega+1)$” denotes the operation of raising a product to the (idempotent) power $\omega + 1$, which in any finite semigroup is well-defined according to $s^\omega$ being the unique idempotent power of $s$.

1. Permutation Identity Type: There exists a nontrivial permutation $\sigma \in S_n$ so that

$$x_1 x_2 \cdots x_n \approx x_{\sigma(1)} x_{\sigma(2)} \cdots x_{\sigma(n)}$$

is satisfied by all members—these are the “permutative” semigroups.

Any nontrivial product identity thus collapses to a specific “regular expansion” or “permutation” form, and all finite semigroups satisfying such an identity fall into maximal pseudovarieties defined by these.

2. Structure and Properties of the Two Families

Expansion-identity pseudovarieties

Semigroups or pseudovarieties defined by expansion identities are structurally “almost completely regular.” In such semigroups, certain high powers or large products of elements necessarily become completely regular (every element is idempotent or invertible in its $\mathcal{D}$-class). Specifically, for

$$x_1\cdots x_n \approx x_1\cdots x_{i-1}(x_i\cdots x_j)^{\omega+1}x_{j+1}\cdots x_n,$$

the induced structural property is that products $S^n$, or more generally certain ideals of the semigroup, are completely regular [(Thumm, 23 Sep 2025), Lemma 4.6]. These maximal pseudovarieties generalize the classically completely regular identity $x \approx x^{\omega+1}$.

Permutation-identity pseudovarieties

Semigroups defined by permutation identities (termed “permutative semigroups”) are those in which the multiplication can be permuted by a fixed nontrivial permutation without changing the outcome for all values. While this generalizes the commutative identity ($xy \approx yx$), not all permutative semigroups are commutative; the permutation can be nontrivial and not generate the full symmetric group.

The crucial point is that, for $V$ a subpseudovariety of $\mathrm{Perm}$ (the pseudovariety of all permutative semigroups), there may or may not exist a common permutation identity that all elements of $V$ satisfy. The obstruction to this arises from the presence of certain nilpotent semigroups.

3. Obstruction by Nilpotent Semigroups and the Role of $T$

A minimal example of a nontrivial nilpotent semigroup is

$T = [x^2 \approx xyx \approx 0]$

meaning both square and triple products vanish. The presence or absence of $T$ in a pseudovariety $V$ of permutative semigroups determines whether $V$ can globally satisfy a fixed permutation identity. The result [(Thumm, 23 Sep 2025), Theorem C] is:

• If $T \nsubseteq V$ and $V \subseteq \mathrm{Perm}$, then $V$ satisfies a common permutation identity.
• If $T \subseteq V$, then $V$ cannot satisfy a nontrivial permutation identity globally.

Thus, $T$ acts as the critical obstruction: its inclusion prevents the existence of a uniform permutation identity for $V$.

4. Maximal Pseudovarieties within the Lattice Structure

The consequences for the lattice of pseudovarieties of finite semigroups are that every family defined by a nontrivial product identity corresponds to the intersection of $\mathrm{Perm}$ with the class of semigroups defined by the corresponding identity, or the class of (almost) completely regular semigroups as defined by the suitable expansion identity. That is, any maximal pseudovariety defined by such identities is determined up to containment by either:

• The maximal expansion-identity pseudovariety for some $n$ and $i \leq j$, or
• The maximal permutation-identity pseudovariety for a given nontrivial permutation $\sigma$.

Obstructions such as $T$ further determine whether the locus of maximals in $\mathrm{Perm}$ is tight (in the sense that inclusion of $T$ collapses all possibility of a uniform permutation identity).

Table: Classification Summary

Identity Type	Maximal Pseudovariety	Structural Implication
Expansion	Defined by $$x_1\cdots x_n \approx x_1\cdots x_{i-1}(x_i\cdots x_j)^{\omega+1}x_{j+1}\cdots x_n$$	$S^n$ or appropriate ideals are completely regular
Permutation	Defined by $$x_1\cdots x_n \approx x_{\sigma(1)}\cdots x_{\sigma(n)}$$	All products invariant under $\sigma$
Obstruction by $T$	Inclusion of $T = [x^2\approx xyx\approx 0]$ rules out any common $\sigma$	No global permutation identity possible if $T\subseteq V$

5. Methods and Inductive Tools

The classification leverages restriction and induction techniques on the arity of product identities. Any nontrivial product identity is shown to imply, via elementary arguments and stabilizations, one of the two canonical forms. In practice, this reduces identity questions for arbitrary product identities either to expansion identities (with associated regularity) or permutation identities (and their corresponding combinatorics), with $T$ dictating when the latter can exist globally within a pseudovariety.

6. Consequences and Applications

This explicit classification resolves outstanding structural ambiguities in the paper of finite semigroup pseudovarieties defined by product identities (Thumm, 23 Sep 2025). The impact is multifold:

• Provides a clear criterion for determining whether a given product identity yields a maximal pseudovariety, and precisely what that maximal class is.
• Establishes the centrality of $T$ as a “forbidden subsemigroup” in the context of permutative semigroup theory, giving a concrete decision process for when global permutation identities exist.
• Supplies technical means for identifying the border between almost completely regular and permutative behavior in the lattice, thereby aiding in the design of equational bases or the analysis of automata and language classes via Eilenberg-type correspondences.

7. Research Directions and Open Problems

The categorization in (Thumm, 23 Sep 2025) suggests further lines of research:

• Investigation of analogous dichotomies for identities involving more complex operations (e.g., higher iterated products, alternating powers).
• Effective determination or enumeration of all maximal pseudovarieties for composites or joins of expansion and permutation identities.
• Deeper paper of the role of nilpotent obstructions (such as $T$) in lattice-theoretic decompositions and their interaction with other natural varieties and pseudovarieties in the sense of spectral theory (Lee et al., 2017).

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The complete classification of maximal pseudovarieties satisfying product identities of the form $x_1\cdots x_n \approx \rho(x_1,\ldots,x_n)$ asserts that every such pseudovariety is, up to intersection with canonical obstructions, completely determined by whether the defining identity collapses to an expansion (almost completely regular) or permutation type, with $T$ as the unique minimal obstruction to globally permutative structure. This result yields a transparent formula for classifying and constructing such maximal pseudovarieties in the lattice of finite semigroups and underpins further structural and algorithmic analysis.