Pseudovarieties Of Semigroups Research Articles

On join irreducible $J$-trivial semigroups

A pseudovariety of semigroups is join irreducible if, whenever it is contained in the complete join of some pseudovarieties, then it is contained in one of the pseudovarieties. A finite semigroup is join irreducible if it generates a join irreducible pseudovariety. New finite $\mathscr{J}$-trivial semigroups $\mathcal{C}\_n$ ($n \geq 2$) are exhibited with the property that, while each $\mathcal{C}\_n$ is not join irreducible, the monoid $\mathcal{C}\_n^I$ is join irreducible. The monoids $\mathcal{C}\_n^I$ are the first examples of join irreducible $\mathscr{J}$-trivial semigroups that generate pseudovarieties that are not self-dual. Several sufficient conditions are also established under which a finite semigroup is not join irreducible. Based on these results, join irreducible pseudovarieties generated by a $\mathscr{J}$-trivial semigroup of order up to six are completely described. It turns out that besides known examples and those generated by $\mathcal{C}\_2^I$ and its dual monoid, there are no further examples.

On Kovács–Newman ordered semigroups

The irreducibility of pseudovarieties of semigroups is studied using many different approaches including a notion of Kovacs–Newman semigroups. In this note, we answer a natural question, whether Kovacs–Newman semigroups can be used in the study of the irreducibility of pseudovarieties of ordered semigroups.

Representations of relatively free profinite semigroups, irreducibility, and order primitivity

We establish that, under certain closure assumptions on a pseudovariety of semigroups, the corresponding relatively free profinite semigroups freely generated by a nonsingleton finite set act faithfully on their minimum ideals. As applications, we enlarge the scope of several previous join irreducibility results for pseudovarieties of semigroups, which turn out to be even join irreducible in the lattice of pseudovarieties of ordered semigroups, so that, in particular, they are not generated by proper subpseudovarieties of ordered semigroups. We also prove the stronger form of join irreducibility for the Krohn-Rhodes complexity pseudovarieties, thereby solving a problem proposed by Rhodes and Steinberg.

A note on the finite basis and finite rank properties for pseudovarieties of semigroups

The finite basis property is often connected with the finite rank property, which it entails. Many examples have been produced of finite rank varieties which are not finitely based. In this note, we establish a result on nilpotent pseudovarieties which yields many similar examples in the realm of pseudovarieties of semigroups.

The κ-word problem over [formula omitted

Profinite techniques are explored in order to prove decidability of a word problem over a family of pseudovarieties of semigroups, which is parameterized by pseudovarieties of groups.Let κ be the signature that naturally generalizes the usual signature on groups: it consists of the multiplication, and of the (ω−1)-power. Given a pseudovariety of groups H, we denote by DRH the pseudovariety all finite semigroups whose regular R-classes lie in H. We prove that the word problem for κ-terms is decidable over DRH provided it is decidable over H (in general, the word problem for κ-terms is said to be decidable over a pseudovariety V if it is decidable whether two κ-terms define the same element in every semigroup of V). Further, we present a canonical form for elements in the free κ-semigroup over DRH, based on the knowledge of a canonical form for elements in the free κ-semigroup over H. This extends work of Almeida and Zeitoun on the pseudovariety of all finite R-trivial semigroups.

On a class of completely join prime J-trivial semigroups with unique involution

A finite algebra is completely join prime if whenever it belongs to the complete join of some collection of pseudovarieties, then it belongs to one of the pseudovarieties. An infinite class of completely join prime J-trivial semigroups with unique involution is introduced to demonstrate the incompatibility between the lattice of pseudovarieties of involution semigroups and the lattice of pseudovarieties of semigroups. Examples are also exhibited to show that a finite involution semigroup and its semigroup reduct need not be simultaneously completely join prime.

Equidivisible pseudovarieties of semigroups

We give a complete characterization of pseudovarieties of semigroups whose finitely generated relatively free profinite semigroups are equidivisible. Besides the pseudovarieties of completely simple semigroups, they are precisely the pseudovarieties that are closed under Mal'cev product on the left by the pseudovariety of locally trivial semigroups. A further characterization which turns out to be instrumental is as the non-completely simple pseudovarieties that are closed under two-sided Karnofsky-Rhodes expansion.

On Pseudovarieties of Forest Algebras

Forest algebras are defined for investigating languages of forests [ordered sequences] of unranked trees, where a node may have more than two [ordered] successors. They consist of two monoids, the horizontal and the vertical, with an action of the vertical monoid on the horizontal monoid, and a complementary axiom of faithfulness. In the study of forest algebras one of the main difficulties is how to handle the faithfulness property. A pseudovariety is a class of finite algebras of a given signature, closed under the taking of homomorphic images, subalgebras and finitary direct products. We tried to adapt in this context some of the results in the theory of semigroups, specially the studies on relatively free profinite semigroups, which are an important tool in the theory of pseudovarieties of semigroups. We define a new version of syntactic congruence of a subset of the free forest algebra, not just a forest language. This new version is the natural extension of the syntactic congruence for monoids in the case of forest algebras and is used in the proof of an analog of Hunter’s Lemma. We show that under a certain assumption the two versions of syntactic congruences coincide. We adapt some results of Almeida on metric semigroups to the context of forest algebras. We show that the analog of Hunter’s Lemma holds for metric forest algebras, which leads to the result that zero-dimensional compact metric forest algebras are residually finite. We show an analog of Reiterman’s Theorem, which is based on a study of the structure profinite forest algebras.

Reducibility vs. definability for pseudovarieties of semigroups

It is easy to show that a pseudovariety which is reducible with respect to an implicit signature [Formula: see text] for the equation [Formula: see text] can also be defined by [Formula: see text]-identities. We present several negative examples for the converse using signatures in which the pseudovarieties are usually defined. An ordered example issue from the extended Straubing–Thérien hierarchy of regular languages is also shown to provide a positive example for the inequality [Formula: see text].

THE CONGRUENCE η* ON SEMIGROUPS

In this paper we define a congruence $\eta^{\ast}$ on semigroups. For the finite semigroups $S$, $\eta^{\ast}$ is the smallest congruence relation such that $S/\eta^{\ast}$ is a nilpotent semigroup (in the sense of Malcev). In order to study the congruence relation $\eta^{\ast}$ on finite semigroups, we define a $\textbf{CS}$-diagonal finite regular Rees matrix semigroup. We prove that, if $S$ is a $\textbf{CS}$-diagonal finite regular Rees matrix semigroup then $S/\eta^{\ast}$ is inverse. Also, if $S$ is a completely regular finite semigroup, then $S/\eta^{\ast}$ is a Clifford semigroup. We show that, for every non-null principal factor $A/B$ of $S$, there is a special principal factor $C/D$ such that every element of $A\setminus B$ is $\eta^{\ast}$-equivalent with some element of $C\setminus D$. We call the principal factor $C/D$, the $\eta^{\ast}$-root of $A/B$. All $\eta^{\ast}$-roots are $\textbf{CS}$-diagonal. If certain elements of $S$ act in the special way on the $\textbf{R}$-classes of a $\textbf{CS}$-diagonal principal factor then it is not an $\eta^{\ast}$-root. Some of these results are also expressed in terms of pseudovarieties of semigroups.

On the irreducibility of pseudovarieties of semigroups

We show that, for every pseudovariety of groups H, the pseudovariety H¯, consisting of all finite semigroups all of whose subgroups lie in H, is irreducible for join and the Mal'cev and semidirect products.

SOME OPERATORS THAT PRESERVE THE LOCALITY OF A PSEUDOVARIETY OF SEMIGROUPS

It is shown that if 𝖵 is a local monoidal pseudovariety of semigroups, then 𝖪 ⓜ 𝖵, 𝖣 ⓜ 𝖵 and [Formula: see text] are local. Other operators of the form 𝖹 ⓜ(_) are considered. In the process, results about the interplay between operators 𝖹 ⓜ(_) and (_) * 𝖣k are obtained.

Implicitly equivalent universal algebras

The concept of implicit operation on pseudovarieties of semigroups goes back to Eilenberg and Schutzenberger [1]. The author in [2–5] generalized this concept to other classes of algebras and established a connection between these operations and positively conditional termal functions in the case of uniform local finiteness of the algebras of the class in question. In this article we put forth the concept of an implicit operation for an arbitrary universal algebra, not necessarily locally finite, and establish a connection between these operations and infinite analogs of positively conditional terms, as well as ∞-quasi-identities arising in the algebraic geometry of universal algebras. We also consider conditions for implicit equivalence of algebras to lattices, semilattices, and Boolean algebras.

Semidirect product with an order-computable pseudovariety and tameness

The semidirect product of pseudovarieties of semigroups with an order-computable pseudovariety is investigated. The essential tool is the natural representation of the corresponding relatively free profinite semigroups and how it transforms implicit signatures. Several results concerning the behavior of the operation with respect to various kinds of tameness properties are obtained as applications.

STABLE PAIRS

For a finite semigroup S and pseudovariety V, (Y, T) is a V-stable pair of S iff Y ⊆ S, T ≤ S and for any relational morphism R : S ⇝ V with V ∈ V there exists a v ∈ V such that Y ⊆ R-1(v) and T ≤ R-1( Stab (v)). X ≤ S is stable if it is generated by an [Formula: see text]-chain {ai} with aiaj = ai for j < i. Given a relation R : S ⇝ A ∈ A (where A denotes the pseudovariety of aperiodic semigroups) that computes PlA(S), we construct a new relation R∞ : S ⇝ (A(M))# that computes A-stable pairs. This proves the main result of this paper: (Y, T) is an A-stable pair of S iff T ≤ ∪ X for some stableX ≤ PlA(S) and Y ⊆ Y' for some Y' ∈ PlA(S) with Y'x = Y' for all x ∈ X. As a corollary we get that if V is a local pseudovariety of semigroups, then V * A has decidable membership problem.

APERIODIC POINTLIKES AND BEYOND

We prove that if π is a recursive set of primes, then pointlike sets are decidable for the pseudovariety of semigroups whose subgroups are π-groups. In particular, when π is the empty set, we obtain Henckell's decidability of aperiodic pointlikes. Our proof, restricted to the case of aperiodic semigroups, is simpler than the original proof.

Characterization of group radicals with an application to Mal’cev products

Radicals for Fitting pseudovarieties of groups are in- vestigated from a profinite viewpoint in order to describe Malcev products on the left by the corresponding local pseudovariety of semigroups.

NORMALLY ORDERED SEMIGROUPS

AbstractIn this paper we introduce the notion of normally ordered block-group as a natural extension of the notion of normally ordered inverse semigroup considered previously by the author. We prove that the class NOS of all normally ordered block-groups forms a pseudovariety of semigroups and, by using the Munn representation of a block-group, we deduce the decompositions in Mal'cev products NOS = EI$\petite m$POI and NOS ∩ A = N$\petite m$POI, where A, EI and N denote the pseudovarieties of all aperiodic semigroups, all semigroups with just one idempotent and all nilpotent semigroups, respectively, and POI denotes the pseudovariety of semigroups generated by all semigroups of injective order-preserving partial transformations on a finite chain. These relations are obtained after showing the equalities BG = EI$\petite m$Ecom = N$\petite m$Ecom, where BG and Ecom denote the pseudovarieties of all block-groups and all semigroups with commuting idempotents, respectively.

Monoids with sub-log-exponential free spectra

A classic result from the 1960s states that the asymptotic growth of the free spectrum of a finite group G is sub-log-exponential if and only if G is nilpotent. Thus a monoid M is sub-log-exponential implies M ∈ G nil ¯ , the pseudovariety of semigroups with nilpotent subgroups. Unfortunately, little more is known about the boundary between the sub-log-exponential and log-exponential monoids. The pseudovariety EDA consists of those finite semigroups satisfying ( x ω y ω ) ω ( y ω x ω ) ω ( x ω y ω ) ω ≈ ( x ω y ω ) ω . Here it is shown that a monoid M is sub-log-exponential implies M ∈ EDA . A quick application: a regular sub-log-exponential monoid is orthodox. It is conjectured that a finite monoid M is sub-log-exponential if and only if it is G nil ¯ ∩ EDA , the finite monoids in EDA having nilpotent subgroups. The forward direction of the conjecture is proved; moreover, the conjecture is proved for S 1 when S is completely (0)-simple. In particular, the six-element Brandt monoid B 2 1 (the Perkins semigroup) is sub-log-exponential.

On Correspondences Between Unary Algebras, Automata, Semigroups and Congruences

In this paper, we give correspondences between unary algebras, semigroups and congruences on free semigroups. We establish isomorphisms between the complete lattice of varieties of semigroups and the complete lattices of families of varieties of unary algebras, and families of filters of congruences on free semigroups. Similar correspondences between generalized varieties and pseudovarieties of semigroups and corresponding families of algebras and congruences are also established.