Pseudovariety of Permutative Semigroups

• Pseudovarieties of permutative semigroups are classes of finite semigroups defined by permutation identities that generalize the commutative law.
• They exhibit structural robustness through operator theory, with permanence operators preserving locality and enabling efficient membership testing.
• Their lattice and epimorphism closure properties offer both theoretical insights and practical algorithms for regular language classification.

A pseudovariety of permutative semigroups is a class of finite semigroups defined by identities in which the order of variables in products is permuted according to a fixed nontrivial permutation pattern. These pseudovarieties generalize the notion of commutative semigroups to encompass broader permutation identities and play a significant role in finite semigroup theory, regular language classification, and algebraic automata theory.

1. Fundamental Definitions and Structural Criteria

A pseudovariety of semigroups is a collection of finite semigroups closed under homomorphic images, subsemigroups, and finite direct products. Permutative semigroups are those satisfying a product identity of the form

$$x_1 x_2 \dots x_n \approx \rho(x_1, x_2, \dots, x_n)$$

where $\rho$ is typically a permutation function, that is, $$\rho(x_1,\dots,x_n) = x_{\sigma(1)} \dots x_{\sigma(n)}$$ for some nontrivial $\sigma \in S_n$ (Thumm, 23 Sep 2025). The classical commutative law $xy \approx yx$ is a basic example.

A key classification result (Thumm, 23 Sep 2025) demonstrates that any nontrivial product identity satisfied by a finite semigroup implies one of two prototype forms:

• An "almost completely regular" expansion:

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

for some $1 \leq i \leq j \leq n$, closely related to block regularity.

• A "pure permutation identity":

$$x_1 \dots x_n \approx x_{\sigma(1)} \dots x_{\sigma(n)}$$

This covers all semigroups in which a common permutation of factors is permitted.

A pseudovariety is called "local" if closure under localization and semidirect product with the delay pseudovariety $D$ coincide: $V = \ell V = V * D$ Operators such as $K(m)$, $D(m)$, and $LI(m)$ preserve locality under mild hypotheses (Costa et al., 2011). If $V$ is defined by permutative identities and is local, then applying such operators yields pseudovarieties with retained locality and permutative structure.

2. Operator Theory: Preservation and Interplay

Natural operators act on pseudovarieties to build more complex classes:

• Semidirect product ($*$), Mal’cev product ($m$), and their variants.
• Permanence operators ($K(m)$, $D(m)$, $LI(m)$) (Costa et al., 2011).

If $V$ is a local monoidal pseudovariety—particularly if defined by permutative-type identities—then $K(m)V$, $D(m)V$, $LI(m)V$, and more generally $Z(m)V$ (where $Z$ is permanent) are local (Costa et al., 2011). For certain $Z$, the equation

$Z@(V*D_k) = (Z@V)*D_k$

holds, demonstrating commutativity of operator application and semidirect product under these conditions.

Permutative pseudovarieties are structurally robust under such operator compositions: locality and permutative behavior are preserved, closure properties for associated language varieties remain intact (e.g., closure under unambiguous concatenation).

3. Characterizations and Membership Algorithms

Some pseudovarieties generated by permutative behavior admit efficient membership testing:

• The power pseudovariety $\mathbf{PCS}$ (generated by all power semigroups of finite completely simple semigroups) can be characterized semantically as aggregates of block groups:

$$S \in \mathbf{PCS} \iff aSb \text{ is a block group for all } a,b\in S$$

Or syntactically by pseudoidentities like

$$(sat)^{\omega}(syt)^{\omega} = (syt)^{\omega}(sxt)^{\omega}$$

Algorithms based on checking block group conditions or ideals are polynomial-time in the semigroup size (Auinger, 2012).

Varieties of normally ordered inverse semigroups (NO), an important class partially defined by permutative–order preserving relations, are generated by monoids of $P$-stable, $P$-order preserving partial permutations on ordered uniform partitions (Caneco et al., 2019). The rank and presentation of such monoids depend only on the number of blocks, facilitating explicit algebraic and combinatorial descriptions.

4. Lattice Properties, Modularity, Cancellability

The lattice of semigroup varieties (SEM) encodes inclusion relationships among different pseudovarieties. For permutational identities of length three:

• Cancellable elements are exactly those expressible as $V = M \vee N$, where $M$ is the trivial or semilattice variety, and $N$ is a nil-variety satisfying all length-3 permutational identities and additional nilpotence $(x^2y \approx 0)$ (Vernikov, 2018).
• Modular elements are less restrictive; there exist modular, non-cancellable permutative varieties.

This rigid containment illustrates that if a variety satisfies one permutational identity, cancellability forces it to satisfy all of a given length.

5. Epimorphism Closure and Structural Robustness

A pseudovariety $V$ is epimorphically closed (or $F$-saturated) if it satisfies strong closure conditions under epimorphisms:

• Every epimorphism to a member of $V$ is onto.
• Every epimorphism from a member of $V$ to a finite semigroup is onto.
• Every finite epimorphic image of a member of $V$ still belongs to $V$ (Almeida et al., 17 Apr 2025).

Permutative pseudovarieties generally satisfy these strong closure properties, provided no non-permutative obstruction is present (e.g., the semigroup $Y$). The symmetric nature ensures that the structure-defining identities survive passage to quotient images.

6. Maximality and Obstruction Results

Maximal pseudovarieties satisfying a product identity are precisely classified (Thumm, 23 Sep 2025):

• If a pseudovariety $V \subseteq \text{Perm}$ does not contain the nilpotency obstruction $T = [x^2 \approx xyx \approx 0]$, it satisfies some permutation identity.
• Conversely, containing $T$ precludes the existence of a common permutation identity across $V$.

This underscores that the only barrier to a universal permutation identity in a permutative pseudovariety is the embedding of nilpotency as in $T$.

7. Applications and Open Problems

Permutative pseudovarieties have direct implications for regular language theory via Eilenberg's correspondence (languages recognized by permutative semigroups possess unique closure properties), decomposition theory, and the computational complexity of word problems (membership, identity checking, pointlikes).

Open questions include:

• The fine structure of the meet-semilattice formed by pseudovarieties defined by distinct block regular or permutation identities,
• The interaction of natural operators with decidability and locality in iterative constructions,
• Further characterizations of permutative behavior within free profinite semigroups, their topological features, and their effects on regular language classification.

Summary Table: Operator Effects on Local Permutative Pseudovarieties

Operator	Effect on Locality	Permutative Structure
K(m), D(m), LI(m)	Preserved (Costa et al., 2011)	Preserved if initial class is permutative
Z(m)(–)	Preserved under permanence (Costa et al., 2011)	Permutative identities stable if Z is permanent
* Dₖ	Locality commutes with operator (Costa et al., 2011)	Structure preserved as long as locality and permanence hold

This synthesis reflects the deep intertwining of algebraic, topological, and computational features of pseudovarieties defined by permutative identities. The framework, including detailed operator theory and maximality criteria, equips researchers with tools for classification, algorithmic analysis, and exploration of advanced questions in semigroup theory and its applications.