Malcev Product Research Articles

关于两个完全单半群的Malcev积的注

关于两个完全单半群的Malcev积的注

On the Malcev products of some classes of epigroups, I

AbstractA semigroup is called an epigroup if some power of each element lies in a subgroup. Under the universal of epigroups, the aim of the paper is devoted to presenting elements in the groupoid together with the multiplication of Malcev products generated by classes of completely simple semigroups, nil-semigroups and semilattices. The information about the set inclusion relations among them is also provided.

Malcev product of superabundant semigroups and its subsemigroups

We use Malcev product of semigroups satisfying some axiomatic conditions to describe the structure of superabundant semigroups and some of its subclasses. Some characterization theorems of these kinds of semigroups are given.

Malcev Products of Monoids and Varieties of Bands

As a generalization of completely regular semigroups, which can be written as \({\mathcal{(G \circ RB) \circ S}}\) where \({\mathcal{G}}\), \({\mathcal{RB}}\) and \({\mathcal S}\) are the varieties of groups, rectangular bands and semilattices, respectively, we have replaced \({\mathcal G}\) by the class \({\mathcal M}\) of monoids. This calls for finding the structure of such semigroups, and, as a first step, characterizations.

Malcev products of weakly cancellative monoids and varieties of bands

We consider unipotent, weakly cancellative, left cancellative, right cancellative, cancellative monoids and groups with the unary operation of mapping onto the identity element and bands with the identity mapping as unary operations. Then form Malcev products of each of the former with varieties of rectangular bands, semilattices, normal bands and all bands. Regarding the resulting semigroups with the induced unary operations, we characterize these classes, some of them we provide with a construction, and determine a basis for their implications, for they all turn out to be quasivarieties. We also determine their relationship which we clarify by an inclusion diagram.

On Malcev Products of Normal Cryptogroups

Normal cryptogroups represented as strong semilattices of completely simple semigroups admit a detailed study of their quasivarieties; in particular of their lattices and groupoids under Malcev product. In this context, we construct the lattice of quasivarieties generated by semilattices \({\mathcal{S}}\), groups \({\mathcal{G}}\), and rectangular bands \({\mathcal{RB}}\) ; it is a 17-element distributive lattice. We compute the groupoids of quasivarieties generated by pairs \({(\mathcal{S}, \mathcal{G}), (\mathcal{S},\mathcal{RB})}\), and \({(\mathcal{G},\mathcal{RB})}\), as well as find the list of quasivarieties which can be expressed by Malcev products with words of length at most equal to 3. This analysis presupposes multiple characterizations of each quasivariety involved in order to establish their relationship, and in particular to prove that all the joins are obtained by means of subdirect products.

Malcev products of unipotent monoids and varieties of bands

Let $\mathcal{U}$ be the class of all unipotent monoids and $\mathcal{B}$ the variety of all bands. We characterize the Malcev product $\mathcal{U} \circ \mathcal{V}$ where $\mathcal{V}$ is a subvariety of $\mathcal{B}$ low in its lattice of subvarieties, $\mathcal{B}$ itself and the subquasivariety $\mathcal{S} \circ \mathcal{RB}$ , where $\mathcal{S}$ stands for semilattices and $\mathcal{RB}$ for rectangular bands, in several ways including by a set of axioms. For members of some of them we describe the structure as well. This succeeds by using the relation $\widetilde{\mathcal{H}}= \widetilde{\mathcal{L}} \cap \widetilde{\mathcal{R}}$ , where $a\;\,\widetilde{\mathcal{L}}\;\,b$ if and only if a and b have the same idempotent right identities, and $\widetilde{\mathcal{R}}$ is its dual. We also consider $(\mathcal{U} \circ \mathcal{RB}) \circ \mathcal{S}$ which provides the motivation for this study since $(\mathcal{G} \circ \mathcal{RB}) \circ \mathcal{S}$ coincides with completely regular semigroups, where $\mathcal{G}$ is the variety of all groups. All this amounts to a generalization of the latter: $\mathcal{U}$ instead of $\mathcal{G}$ .

On the complexity of computing the profinite closure of a rational language

Profinite topology is used in the classification of rational languages. In particular, several important decidability problems, related to the Malcev product, reduce to the computation of the closure of a rational language in the profinite topology. It is known that, given a rational language by a deterministic automaton, computing a deterministic automaton accepting its profinite closure can be done with an exponential upper bound. This paper is dedicated the study of a lower bound for this problem: we prove that, in some cases, if the alphabet contains at least three letters, it requires an exponential time.

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.

Semigroup expansions using the derived category, kernel, and Malcev products

Three new classes of expansions are defined in this paper. More precisely, three different expansions are associated to each semigroup variety V. It is shown that several previously defined expansions can be viewed as specific examples of these constructions, or slight variants there of. This method is then used to “smooth” an already existing expansion to one which is guaranteed to be functorial and is maximal in a sense that will be made precise. Perhaps more importantly, this method of construction provides a large resource of expansions to be used as needed in the future.

INVERSE MONOIDS OF DOT-DEPTH TWO

It is shown that the pseudovariety of inverse monoids of dot-depth two is decidable and is equal to the Malcev product of the pseudovariety generated by the 5 element Brandt semigroup and the variety of semilattices.

THE BIDETERMINISTIC CONCATENATION PRODUCT

This paper is devoted to the study of the bideterministic concatenation product, a variant of the concatenation product. We give an algebraic characterization of the varieties of languages closed under this product. More precisely, let V be a variety of monoids, [Formula: see text] the corresponding variety of languages and [Formula: see text] the smallest variety containing [Formula: see text] and the bideterministic products of two languages of [Formula: see text]. We give an algebraic description of the variety of monoids [Formula: see text] corresponding to [Formula: see text]. For instance, we compute [Formula: see text] when V is one of the following varieties: the variety of idempotent and commutative monoids, the variety of monoids which are semilattices of groups of a given variety of groups, the variety of ℛ-trivial and idemptotent monoids. In particular, we show that the smallest variety of languages closed under bideterministic product and containing the language {1}, corresponds to the variety of [Formula: see text]-trivial monoids with commuting idempotents. Similar results were known for the other variants of the concatenation product, but the corresponding algebraic operations on varieties of monoids were based on variants of the semidirect product and of the Malcev product. Here the operation [Formula: see text] makes use of a construction which associates to any finite monoid M an expansion [Formula: see text] with the following properties: (1) M is a quotient of [Formula: see text] (2) the morphism [Formula: see text] induces an isomorphism between the submonoids of [Formula: see text] and of M generated by the regular elements and (3) the inverse image under π of an idempotent of M is a 2-nilpotent semigroup.

ASH'S TYPE II THEOREM, PROFINITE TOPOLOGY AND MALCEV PRODUCTS: PART I

This paper is concerned with the many deep and far reaching consequences of Ash's positive solution of the type II conjecture for finite monoids. After reviewing the statement and history of the problem, we show how it can be used to decide if a finite monoid is in the variety generated by the Malcev product of a given variety and the variety of groups. Many interesting varieties of finite monoids have such a description including the variety generated by inverse monoids, orthodox monoids and solid monoids. A fascinating case is that of block groups. A block group is a monoid such that every element has at most one semigroup inverse. As a consequence of the cover conjecture — also verified by Ash — it follows that block groups are precisely the divisors of power monoids of finite groups. The proof of this last fact uses earlier results of the authors and the deepest tools and results from global semigroup theory. We next give connections with the profinite group topologies on finitely generated free monoids and free groups. In particular, we show that the type II conjecture is equivalent with two other conjectures on the structure of closed sets (one conjecture for the free monoid and another one for the free group). Now Ash's theorem implies that the two topological conjectures are true and independently, a direct proof of the topological conjecture for the free group has been recently obtained by Ribes and Zalesskii. An important consequence is that a rational subset of a finitely generated free group G is closed in the profinite topology if and only if it is a finite union of sets of the form gH1H2…Hn, where g ∈ G and each Hi is a finitely generated subgroup of G. This significantly extends classical results of M. Hall. Finally, we return to the roots of this problem and give connections with the complexity theory of finite semigroups. We show that the largest local complexity function in the sense of Rhodes and Tilson is computable.

The Malcev product of quasivarieties of normal bands

The Malcev product of quasivarieties of normal bands