In this paper we develop an ideal structure theory for the class of left reductive regular semigroups and apply it to several subclasses of popular interest. In these classes we observe that the right ideal structure of the semigroup is 'embedded' inside the left ideal one, and so we can construct these semigroups starting with only one object (unlike in other more general cases). To this end, we introduce an upgraded version of Nambooripad's normal category as our building block. We dub these new categories as connected categories. The main theorem of the paper describes a category equivalence between the category of left reductive regular semigroups and the category of connected categories. Then, we specialise our result to describe constructions of L-unipotent semigroups, right regular bands, inverse semigroups and arbitrary regular monoids. Intriguingly, the same results hold for the dual classes of right reductive regular semigroups, R-unipotent semigroups and left regular bands; this is due to the category isomorphism between the left and right reductive regular semigroups. Finally, we provide concrete and relatively simple descriptions of the connected categories that arise from finite transformation semigroups, linear transformation semi groups (over a finite dimensional vector space) and symmetric inverse monoids. (c) 2026 The Author(s). Published by Elsevier Inc. This is an open access article under the CC BY license (http:// creativecommons.org/licenses/by/4.0/).

In group theory, Fitting-formations are important classes and object of large study. For inverse semigroups, i-formations and i-Fitting classes have been considered, extending the group case. They are related to correspondences of idempotent separating congruences (and of languages contained in the centralisers). The aim of this paper is to characterise the correspondences of congruences (and of languages) associated to i-Fitting-formations of inverse semigroups. Although the known definitions of i-formation and i-Fitting class of congruences (of languages) are natural and appear in some way as dual of each other, they give rise to classes that intersect trivially, thence that are not associated to i-Fitting-formations of inverse semigroups. This led to the search for different correspondences, denoted by i-Fitting systems. The intersection of the class of all i-Fitting systems with the class of all i-formations of congruences (of languages) is then in bijection with the class of all i-Fitting-formations of inverse semigroups. In the case of Clifford semigroups (in particular, of groups), a dual path can be taken and another kind of correspondence of congruences (of languages) is obtained, which is also in bijecton with Fitting-formations of Clifford semigroups (groups).

This article starts by relating formations of inverse semigroups, of congruences on inverse semigroups and of congruences on free inverse semigroups. Then, i-formations of idempotent separating congruences and of languages are defined and their classes are proved to be in bijection, and in bijection with f-formations of inverse semigroups. In the second part, the concepts of i-Fitting class of inverse semigroups, of congruences and of languages are introduced. Bijections between s-fitting classes of inverse semigroups, i-fitting classes of congruences and i-fitting classes of languages are presented. Of special relevance are results on formations and Fitting classes induced by the inverse case on the group case.

Taking formations of groups and of inverse semigroups as the starting point, formations of orthodox semigroups are defined, as well as the wider class of i-formations (i standing for idempotent-separating). The relation between the nature of a class of inverse semigroups ℱ [of groups 𝒢 ] and that of certain classes of orthodox semigroups with associated inverse semigroups in ℱ [groups in 𝒢 ] is discussed. The product of formations of orthodox semigroups, in particular of R -unipotent semigroups, is considered, and a product like the Gaschütz product known for groups is presented for i-formations. The paper concludes with a list of questions.

This article explores a generalisation of the theory of formations of groups. Taking formations of groups as the starting point, formations of inverse semigroups are defined, as well as the wider classes of i-formations (i standing for idempotent-separating) and some classes of the kind named f-formations (f standing for fundamental). The relation between the nature of a class of groups and that of certain classes of inverse semigroups with associated groups in the first is discussed. The product of formations is considered, and a product like the Gaschütz’s product known for groups is presented for f-formations.

This article explores a generalization of the algebraic theory of formal languages. Having, as starting point, the work of T. Colcombet on cost functions and stabilization monoids, and of Daviaud et al. on stabilization algebras, this class of algebras is extended to omega#-algebras and omega#-automata are also introduced. The equality problem for order ideals (of free omega#-algebras) recognized by finite omega#-algebras is answered positively in this context. Various results on formal languages and monoids are generalized to this setting of order ideals and omega#-algebras. The class of cost functions is proved to be embeddable in the class of recognizable order ideals.

This article explores a generalization of the algebraic theory of formal languages. Having, as starting point, the work of T. Colcombet on cost functions and stabilization monoids, and of Daviaud et al. on stabilization algebras, this class of algebras is extended to omega#-algebras and omega#-automata are also introduced. The equality problem for order ideals (of free omega#-algebras) recognized by finite omega#-algebras is answered positively in this context. Various results on formal languages and monoids are generalized to this setting of order ideals and omega#-algebras. The class of cost functions is proved to be embeddable in the class of recognizable order ideals.

Abstract This article explores a generalization of the algebraic theory of formal languages. Having, as starting point, the work of T. Colcombet on cost functions and stabilization monoids, and of Daviaud et al. on stabilization algebras, this class of algebras is extended to ω♯-algebras and ω♯-automata are also introduced. The equality problem for order ideals (of free ω♯-algebras) recognized by finite ω♯-algebras is answered positively in this context. Various results on formal languages and monoids are generalized to this setting of order ideals and ω♯-algebras. The class of cost functions is proved to be embeddable in the class of recognizable order ideals.

Topological procedures to relate pseudoinequalities that define a pseudovariety of ordered algebras with inequalities that ultimately define it, and vice-versa, are presented.

A formation of monoids is a class of finite monoids closed under taking quotients and subdirect products. Formations of monoids were first studied in connection with formal language theory, but in this paper, we come back to an algebraic point of view. We give two natural constructions of formations based on constraints on the minimal ideal and on the maximal subgroups of a monoid. Next we describe two sublattices of the lattice of all formations, and give, for each of them, an isomorphism with a known lattice of varieties of monoids. Finally, we study formations and varieties containing only Clifford monoids, completely describe such varieties and discuss the case of formations.

For an arbitrary group G, it is known that either the semigroup rank \(G{\text {rk}_\text {s}}\) equals the group rank \(G{\text {rk}_\text {g}}\), or \(G{\text {rk}_\text {s}}= G{\text {rk}_\text {g}}+1\). This is the starting point for the research of the article, where the precise relation between both ranks for diverse kinds of groups is established. The semigroup rank of any relatively free group is computed. For a finitely generated abelian group G, it is proved that \(G{\text {rk}_\text {s}}= G{\text {rk}_\text {g}}+1\) if and only if G is torsion-free. In general, this is not true. Partial results are obtained in the nilpotent case. It is also shown that if M is a connected closed surface, then \((\pi _1(M)){\text {rk}_\text {s}}= (\pi _1(M)){\text {rk}_\text {g}}+1\) if and only if M is orientable.

It is known that an Ehresmann monoid P(T, Y) may be constructed from a monoid T acting via order-preserving maps on both sides of a semilattice Y with identity, such that the actions satisfy an appropriate compatibility criterion. Our main result shows that if T is cancellative and equidivisible (as is the case for the free monoid X*), the monoid P(T, Y) not only is Ehresmann but also satisfies the stronger property of being adequate. Fixing T, Y and the actions, we characterise P(T, Y) as being unique in the sense that it is the initial object in a suitable category of Ehresmann monoids. We also prove that the operator P defines an expansion of Ehresmann monoids. (C) 2018 Elsevier Inc. All rights reserved.

Malcev described the congruences of the monoid $T_n$ of all full transformations on a finite set $X_n=\{1, \dots,n\}$. Since then, congruences have been characterized in various other monoids of (partial) transformations on $X_n$, such as the symmetric inverse monoid $In_n$ of all injective partial transformations, or the monoid $PT_n$ of all partial transformations. The first aim of this paper is to describe the congruences of the direct products $Q_m\times P_n$, where $Q$ and $P$ belong to $\{T, PT,In\}$. Malcev also provided a similar description of the congruences on the multiplicative monoid $F_n$ of all $n\times n$ matrices with entries in a field $F$, our second aim is provide a description of the principal congruences of $F_m \times F_n$. The paper finishes with some comments on the congruences of products of more than two transformation semigroups, and a fairly large number of open problems.

Takahasi's Theorem on chains of subgroups of bounded rank in a free group is generalized to several classes of semigroups. As an application, it is proved that the subsemigroups of periodic points are finitely generated and periodic orbits are bounded for arbitrary endomorphisms for various semigroups. Some of these results feature classes such as completely simple semigroups, Clifford semigroups or monoids defined by balanced one-relator presentations.

In this paper we investigate some classes of semigroup rings with respect to (semi)primeness and (semi)primitivity. We do so by extending the techniques developed by Munn in (Proc R Soc Edinbur Sect A 107:175–196, 1987 ) and (Proc R Soc Edinbur Sect A 115:109–117, 1990 ) for the study of semigroup rings of inverse semigroups. Restriction, weakly ample and ample semigroups are considered.

Ehresmann monoids form a variety of bi-unary monoids, that is, monoids equipped with two basic unary operations, the images of which coincide and form a semilattice of projections. The monoid of binary relations B-X on any set X with unary operations of domain and range is Ehresmann. Inverse monoids, regarded as bi-unary submonoids of B-X via the Wagner Preston representation theorem, are therefore also Ehresmann. At the other extreme, any monoid is Ehresmann, where the unary operations take all elements to the monoid identity. We demonstrate here using semilattices and monoids as building blocks that Ehresmann monoids have a rich structure, fundamentally different from that of inverse monoids and, indeed, from that of the interim class of restriction raonoids.The article introduces a notion of properness for Ehresmann monoids, that tightly controls structure and is dependent upon sets of generators. We show how to construct an Ehresmann monoid P(T, Y) satisfying our properness condition from a semilattice Y acted upon on both sides by a monoid T via order preserving maps. The free Ehresmann monoid on X is proven to be of the form P(X*, Y). The next question deals with the existence of proper covers. We answer it in a positive way, proving that any Ehresmann monoid M admits a cover of the form P(X*,E), where E is the semilattice of projections of M. Here a 'cover' is a preimage under a morphism that separates elements in E. (C) 2015 Elsevier Inc. All rights reserved.

In this note we consider various classes of monoids of transformations on a finite chain, in particular of transformations that preserve or reverse either the order or the orientation. Being finite monoids we are naturally interested in computing both their cardinals and their idempotent numbers. Fibonacci and Lucas numbers play an essential role in the last computations.