Cev Products Research Articles

About the power pseudovariety PCS

The power pseudovariety PCS, that is, the pseudovariety of finite semigroups generated by all power semigroups of finite completely simple semigroups has recently been characterized as the pseudovariety AgBG of all so-called aggregates of block groups. This characterization can be expressed as the equality of pseudovarieties PCS=AgBG. In fact, a longer sequence of equalities of pseudovarieties, namely the sequence of equalities PCS=J∗CS=JⓜCS=AgBG has been verified at the same time. Here, J is the pseudovariety of all 𝒥-trivial semigroups, CS is the pseudovariety of all completely simple semigroups, J∗CS is the pseudovariety generated by the family of all semidirect products of 𝒥-trivial semigroups by completely simple semigroups, and JⓜCS is the pseudovariety generated by the Mal’cev product of the pseudovarieties J and CS. In this paper, another different proof of these equalities is provided first. More precisely, the equalities PCS=J∗CS=JⓜCS are given a new proof, while the equality JⓜCS=AgBG is quoted from a foregoing paper. Subsequently in this paper, this new proof of the mentioned equalities is further refined to yield a proof of the following more general result: For any pseudovariety H of groups, let CS(H) stand for the pseudovariety of all completely simple semigroups whose subgroups belong to H. Then it turns out that, for every locally extensible pseudovariety H of groups, the equalities of pseudovarieties P(CS(H))=J∗CS(H)=JⓜCS(H) hold.

Semiring varieties related to multiplicative Green’s relations on a semiring

In this paper, by means of congruence openings of multiplicative Green’s relations on a semiring we define and study several varieties of semirings, obtain the relationship between these varieties and give Mal’cev product decompositions of some varieties of idempotent semirings. In particular, we establish order embeddings of the lattice of all subvarieties of the variety of multiplicatively idempotent semirings (resp. idempotent semirings) into the direct products of the lattices of open (resp. closed) varieties with respect to two opening (resp. closure) operators on this lattice what we introduced.

Nilpotency and strong nilpotency for finite semigroups

AbstractNilpotent semigroups in the sense of Mal’cev are defined by semigroup identities. Finite nilpotent semigroups constitute a pseudovariety, MN, which has finite rank. The semigroup identities that define nilpotent semigroups lead us to define strongly Mal’cev nilpotent semigroups. Finite strongly Mal’cev nilpotent semigroups constitute a non-finite rank pseudovariety, SMN. The pseudovariety SMN is strictly contained in the pseudovariety MN, but all finite nilpotent groups are in SMN. We show that the pseudovariety MN is the intersection of the pseudovariety BGnil with a pseudovariety defined by a κ-identity. We further compare the pseudovarieties MN and SMN with the Mal’cev product 𝖩ⓜ𝖦nill.

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 ITERATED MAL'CEV PRODUCTS WITH A PSEUDOVARIETY OF GROUPS

We characterize H-solvable semigroups with respect to a pseudovariety H of groups. Applications are given to iterated Mal'cev products with H.

An upper bound for the power pseudovariety PCS

It is a celebrated result in finite semigroup theory that the equality of pseudovarieties PG = BG holds, where PG is the pseudovariety of finite monoids generated by all power monoids of finite groups and BG is the pseudovariety of all block groups, that is, the pseudovariety of all finite monoids all of whose regular \({\fancyscript{D}}\)-classes have the property that the corresponding principal factors are inverse semigroups. Moreover, it is well known that BG = JⓜG, where JⓜG is the pseudovariety of finite monoids generated by the Mal’cev product of the pseudovarieties J and G of all finite \({\fancyscript{J}}\)-trivial monoids and of all finite groups, respectively. In this paper, a more general kind of finite semigroups is considered; namely, the so-called aggregates of block groups are introduced. It follows that the class AgBG of all aggregates of block groups forms a pseudovariety of finite semigroups. It is next proved that AgBG = JⓜCS, where JⓜCS is the pseudovariety of finite semigroups generated by the Mal’cev product of the pseudovarieties J and CS, whilst, this once, J stands for the pseudovariety of all finite \({\fancyscript{J}}\)-trivial semigroups and CS stands for the pseudovariety of all finite completely simple semigroups. Furthermore, it is shown that the power pseudovariety PCS, which is the pseudovariety of finite semigroups generated by all power semigroups of finite completely simple semigroups, has the property that \({{\bf PCS}\subseteq{\bf AgBG}}\). However, the question whether this inclusion is strict or not is left open.

On the lattice of sub-pseudovarieties of DA

The wealth of information that is available on the lattice of varieties of bands, is used to illuminate the structure of the lattice of sub-pseudovarieties of DA, a natural generalization of bands which plays an important role in language theory and in logic. The main result describes a hierarchy of decidable sub-pseudovarieties of DA in terms of iterated Mal’cev products with the pseudovarieties of definite and reverse definite semigroups.

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.

Some results onC-varieties

In an earlier paper, the second author generalized Eilenberg's variety theory by establishing a basic correspondence between certain classes of monoid morphisms and families of regular languages. We extend this theory in several directions. First, we prove a version of Reiterman's theorem concerning the definition of varieties by identities, and illustrate this result by describing the identities associated with languages of the form (a_1a_2... a_k)+, where a_1, ..., a_k are distinct letters. Next, we generalize the notions of Mal'cev product, positive varieties, and polynomial closure. Our results not only extend those already known, but permit a unified approach of different cases that previously required separate treatment.

ON SEMIGROUPS WHOSE IDEMPOTENT-GENERATED SUBSEMIGROUP IS APERIODIC

We show that if S is a finite semigroup with aperiodic idempotent-generated subsemigroup and H is a pseudovariety of groups, then the sequence of iterated H-kernels of S stops at the idempotent-generated subsemigroup if and only if each subgroup of S belongs to the wreath product closure of H. Applications are given to Mal'cev products.

A conjecture on the concatenation product

In a previous paper, the authors studied the polynomial closure of a variety of languages and gave an algebraic counterpart, in terms of Mal'cev products, of this operation. They also formulated a conjecture about the algebraic counterpart of the boolean closure of the polynomial closure – this operation corresponds to passing to the upper level in any concatenation hierarchy. Although this conjecture is probably true in some particular cases, we give a counterexample in the general case. Another counterexample, of a different nature, was independently given recently by Steinberg. Taking these two counterexamples into account, we propose a modified version of our conjecture and some supporting evidence for that new formulation. We show in particular that a solution to our new conjecture would give a solution of the decidability of the levels 2 of the Straubing–Thérien hierarchy and of the dot-depth hierarchy. Consequences for the other levels are also discussed.

Some pseudovariety joins involvinglocally trivial semigroups

This paper is concerned with the computation of pseudovariety joins involving the pseudovariety L I of locally trivial semigroups. We compute, in particular, the join of L I with any subpseudovariety of CR(m in circle)N, the Mal’cev product of the pseudovariety of completely regular semigroups and the pseudovariety of nilpotent semigroups. Similar studies are conducted for the pseudovarieties K, D and N, where K (resp. D) is the pseudovariety of all semigroups S such that eS=e (resp. Se=e ) for each idempotent e of S .

Hyperdecidability of pseudovarieties of orthogroups

Let W denote the intersection with the pseudovariety of completely regular semigroups of the Mal'cev product of the pseudovariety of bands with a pseudovariety V of completely regular semigroups. It is shown that the (pseudo)word problem for W is reduced to that for V in such a way that decidability is preserved in the case in which terms involving only multiplication and weak inversion are considered. It is also shown that, if V is a hyperdecidable (respectively canonically reducible) pseudovariety of groups, then so is W.

Abelian Kernels of Some Monoids of Injective Partial Transformations and an Application

In this paper we compute the abelian kernels of the monoids POI n and POPI n of all injective order preserving and respectively, orientation preserving, partial transformations on a chain with n elements. As an application, we show that the pseudovariety POPI generated by the monoids POPI n (n epsilon N) is not contained in the Mal'cev product of the pseudovariety POI generated by the monoids POI n (n epsilon N) with the pseudovariety Ab of all finite abelian groups.

Fundamental groups,inverse schützenberger automata,and monoid presentations

This paper gives decidable conditions for when a finitely generated subgroup of a free group is the fundamental group of a Schützenberger automaton corresponding to a monoid presentation of an inverse monoid. Also, generalizations are given to specific types of inverse monoids as well as to monoids which are "nearly inverse." This result has applications to computing membership for inverse monoids in a Mal'cev product of the pseudovariety of semilattices with a pseudovariety of groups. This paper also shows that there is a bijection between strongly connected inverse automata and subgroups of a free group, generated by positive words. Hence, we also obtain that it is decidable whether a finite strongly connected inverse automaton is a Schützenberger automaton corresponding to a monoid presentation of an inverse monoid. Again, we have generalizations to other types of inverse monoids and to "nearly inverse" monoids. We show that it is undecidable whether a finite strongly connected inverse automaton is a Schützenberger automaton of a monoid presentation of anE-unitary inverse monoid.

Associativity of products of existence varieties of regular semigroups

An e-variety is a class of regular semigroups that is closed under the formation of direct products, homomorphic images and regular subsemigroups. In a previous paper, the authors established that, for any nongroup e-variety V , the e-variety LG o V , where o denotes the Mal'cev product within the class of regular semigroups and LG denotes the e-variety of left groups, is actually equal to the e-variety G ⊗ V generated by all wreath products of the form G ⊗ T, where G ϵ G , the e-variety of all groups, and T ϵ V . It was also shown that if LF denotes the e-variety of left zero semigroups and J the e-variety of all semilattices, then LF o V is equal to the e-variety J ⊗ ∗ V generated by certain subsemigroups of the wreath products of the form S ⊗ T, where S ϵ J and T ϵ V . In this paper, the e-variety generated by the regular parts of the wreath products of the form RF ⊗ V , RB ⊗ V and BJ ⊗ V , where RF , RB and BJ denote the e-variety of right zero semigroups, rectangular bands and completely simple semigroups respectively, are studied and are found, in general, to fall far short of the corresponding Mal'cev products. An important tool is the associativity of the wreath product of e-variety under certain conditions and a substantial part of the paper is devoted to this issue.

Irreducibility of certain pseudovarieties1

We prove that the pseudovarieties of all finite semigroups, and of all aperiodic finii e semigroups are irreducible for join, for semidirect product and for Mal’cev product. In particular, these pseudovarieties do not admit maximal proper subpseudovarieties. More generally, analogous results are proved for the pseudovariety of all finite semigroups all of whose subgroups are in a fixed pseudovariety of groups H, provided th.it H is closed under semidirect product.

Bivarieties of Orthodox Semigroups Generated by Generalized Mal'cev Products

Bivarieties of Orthodox Semigroups Generated by Generalized Mal'cev Products

Polynomial closure and unambiguous product

This article is a contribution to the algebraic theory of automata, but it also contains an application to Buchi’s sequential calculus. The polynomial closure of a class of languagesC is the set of languages that are finite unions of languages of the formL 0 a 1 L 1 ···a nLn, where thea i’s are letters and theL i’s are elements ofC. Our main result is an algebraic characterization, via the syntactic monoid, of the polynomial closure of a variety of languages. We show that the algebraic operation corresponding to the polynomial closure is a certain Mal’cev product of varieties. This result has several consequences. We first study the concatenation hierarchies similar to the dot-depth hierarchy, obtained by counting the number of alternations between boolean operations and concatenation. For instance, we show that level 3/2 of the Straubing hierarchy is decidable and we give a simplified proof of the partial result of Cowan on level 2. We propose a general conjecture for these hierarchies. We also show that if a language and its complement are in the polynomial closure of a variety of languages, then this language can be written as a disjoint union of marked unambiguous products of languages of the variety. This allows us to extend the results of Thomas on quantifier hierarchies of first-order logic.