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

Mal $$'$$ cev described the congruences of the monoid $$\mathcal {T}_n$$ 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 $$\mathcal {I}_n$$ of all injective partial transformations, or the monoid $$\mathcal {PT}_n$$ 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 $$\{\mathcal {T}, \mathcal {PT},\mathcal {I}\}$$ . Mal $$'$$ cev 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 to 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 on a number of related open problems.

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. The first and second authors received financial support from Fundação para a Ciência e a Tecnologia (FCT) through the following four projects: UID/MULTI/04621/2013, UIDB/04621/2020 and UIDP/04621/2020 of CEMAT at Faculdade de Ciências, Universidade de Lisboa, and project PTDC/MAT-PUR/31174/2017. The third author is partially funded by the DeLTA project (ANR-16-CE40-0007).

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.

Submitted by Fátima Piedade (fpiedade@sa.isel.pt) on 2016-04-15T16:35:26Z No. of bitstreams: 1 The semigroup ring of a restriction semigroup with an inverse skeleton.pdf: 282206 bytes, checksum: bca554f5eb8203ea098ae18b46217f63 (MD5) Made available in DSpace on 2016-04-15T16:35:26Z (GMT). No. of bitstreams: 1 The semigroup ring of a restriction semigroup with an inverse skeleton.pdf: 282206 bytes, checksum: bca554f5eb8203ea098ae18b46217f63 (MD5) Previous issue date: 2015-04

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.

This is the first of two articles studying the structure of left adequate and, more generally, of left Ehresmann monoids. Motivated by a careful analysis of normal forms, we introduce here a concept of proper for a left adequate monoid M. In fact, our notion is that of T-proper, where T is a submonoid of M. We show that any left adequate monoid M has an X*-proper cover for some set X, that is, there is a left adequate monoid [Formula: see text] that is X*-proper, and an idempotent separating surjective morphism [Formula: see text] of the appropriate type. Given this result, we may deduce that the free left adequate monoid on any set X is X*-proper. In a subsequent paper, we show how to construct T-proper left adequate monoids from any monoid T acting via order-preserving maps on a semilattice with identity, and prove that the free left adequate monoid is of this form. An alternative description of the free left adequate monoid will appear in a paper of Kambites. We show how to obtain the labeled trees appearing in his result from our structure theorem. Our results apply to the wider class of left Ehresmann monoids, and we give them in full generality. We also indicate how to obtain some of the analogous results in the two-sided case. This paper and its sequel, and the two of Kambites on free (left) adequate semigroups, demonstrate the rich but accessible structure of (left) adequate semigroups and monoids, introduced with startling insight by Fountain some 30 years ago.

Proper extensions that are “injective on ℒ-related idempotents” of ℛ-unipotent semigroups, and much more generally of the class of generalised left restriction semigroups possessing the ample and congruence conditions, referred to here as glrac semigroups, are described as certain subalgebras of a λ-semidirect product of a left regular band by an ℛ-unipotent or by a glrac semigroup, respectively. An example of such is the generalized Szendrei expansion. As a consequence of our embedding, we are able to give a structure theorem for proper left restriction semigroups. Further, we show that any glrac semigroup S has a proper cover that is a semidirect product of a left regular band by a monoid, and if S is left restriction, the left regular band may be taken to be a semilattice.

We show that the free weakly E-ample monoid on a set X is a full submonoid of the free inverse monoid FIM(X) on X. Consequently, it is ample, and so coincides with both the free weakly ample and the free ample monoid FAM(X) on X. We introduce the notion of a semidirect product Y*T of a monoid T acting doubly on a semilattice Y with identity. We argue that the free monoid X* acts doubly on the semilattice [Formula: see text] of idempotents of FIM(X) and that FAM(X) is embedded in [Formula: see text]. Finally we show that every weakly E-ample monoid has a proper ample cover.

The generalized Szendrei expansion SPr is defined for an \( \mathcal{R} \)-unipotent semigroup S and the projection of SPr onto S is proved to be an initial object in the category of all F-morphisms onto S. Also, the languages recognized by these expansions are described following the semidirect product principle.

The construction by Hall of a fundamental orthodox semigroup WB from a band B provides an important tool in the study of orthodox semigroups. We present here a semigroup SB that plays the role of WB for a class of semigroups having a band of idempotents B. Specifically, the semigroups we consider are weakly B-abundant and satisfy the congruence condition (C). Any orthodox semigroup S with E(S)=B lies in our class. On the other hand, if a semigroup S lies in our class, then S is Ehresmann if and only if B is a semilattice.

An appropriate generalization of the notion of permissible sets of inverse semigroups is found within the class of weakly ample semigroups that allows us to introduce the notion of an almost left factorizable weakly ample semigroup in a way analogous to the inverse case. The class of almost left factorizable weakly ample semigroups is proved to coincide with the class of all (idempotent separating) (2, 1, 1)-homomorphic images of semigroups W(T, Y) where Y is a semilattice, T is a unipotent monoid acting on Y, and W(T, Y) is a well-defined subsemigroup in the respective semidirect product that appeared in the structure theory of left ample monoids more than ten years ago. Moreover, the semigroups W(T, Y) are characterized to be, up to isomorphism, just the proper and almost left factorizable weakly ample semigroups.

In this paper we describe the covers of a left ample semigroup that arise from strict (2,1)-embeddings in left factorizable left ample monoids.

Proper ample monoids are described by means of a certain category acted upon on both sides by a cancellative monoid. Making use of this characterization, we show that every ample monoid $S$ has a proper ample cover, which can be taken to be finite whenever $S$ is finite.

The generalized prefix expansion of inverse semigroups, presented by Lawson, Margolis and Steinberg, is suitably modified to define an expansion for weakly left ample semigroups. We consider the class of FA-morphisms between weakly left ample semigroups and show that this expansion gives rise to a universal FA-morphism onto a weakly left ample semigroup. The methods used are necessarily different from the ones applied in the inverse case. We show how the inverse case can be deduced from this more general situation.

We consider proper (idempotent pure) extensions of weakly left ample semigroups. These are extensions that are injective in each ]]> \widetilde{\mathcal{R}}$-class. A graph expansion of a weakly left ample semigroup S is shown to be such an extension of S. Using semigroupoids acted upon by weakly left ample semigroups, we prove that any weakly left ample semigroup which is a proper extension of another such semigroup T is (2,1)-embeddable into a λ-semidirect product of a semilattice by T. Some known results, by O'Carroll, for idempotent pure extensions of inverse semigroups and, by Billhardt, for proper extensions of left ample semigroups follow from this more general situation.

Glasgow Mathematical Journal, nº 47 (2005), pg. 413-424 This paper is mainly dedicated to describing the congruences on certain monoids of transformations on a finite chain Xn with n elements. Namely, we consider the monoids ODn and PODn of all full, respectively partial, transformations on Xn that preserve or reverse the order, as well as the submonoid POn of PODn of all its order-preserving elements. The inverse monoid PODIn of all injective elements of PODn is also considered. We show that in POn any congruence is a Rees congruence, but this may not happen in the monoids ODn, PODIn and PODn. However in all these cases the congruences form a chain. This work was developed within the activities of Centro de ´Algebra da Universidade de Lisboa, supported by FCT and FEDER, within project POCTI ”Fundamental and Applied Algebra”

In this paper we calculate presentations for some natural monoids of transformations on a chain X n = {1

Weakly left ample semigroups are a class of semigroups that are (2,1)-subalgebras of semigroups of partial transformations, where the unary operation takes a transformation α to the identity map in the domain of α. It is known that there is a class of proper weakly left ample semigroups whose structure is determined by unipotent monoids acting on semilattices or categories. In this paper we show that for every finite weakly left ample semigroup S, there is a finite proper weakly left ample semigroup Ŝ and an onto morphism from Ŝ to S which separates idempotents. In fact, Ŝ is actually a (2,1)-subalgebra of a symmetric inverse semigroup, that is, it is a left ample semigroup (formerly, left type A).

The celebrated construction by Munn of a fundamental inverse semigroup TE from a semilattice E provides an important tool in the study of inverse semigroups. We present here a semigroup CE that plays the TE role for Ehresmann semigroups. Inverse semigroups are Ehresmann, as are those that are adequate, weakly ample or weakly hedged. We describe explicitly the semigroups CE for some specific semilattices E and extract information relating to the corresponding classes of Ehresmann semigroups.

Finite generation and presentability of general unions of semigroups, as well as of bands of semigroups, bands of monoids, semilattices of semigroups and strong semilattices of semigroups, are investigated. For instance, it is proved that a band Y of monoids S α (α∈ Y ) is finitely generated/presented if and only if Y is finite and all S α are finitely generated/presented. By way of contrast, an example is exhibited of a finitely generated semigroup which is not finitely presented, but which is a disjoint union of two finitely presented subsemigroups.

It is well known that the free group on a non-empty set can be totally ordered and, further, that each compatible latttice ordering on a free group is a total ordering. On the other hand, Saitô has shown that no non-trivial free inverse semigroup can be totally ordered. In this note we show, however, that every free inverse monoid admits compatible lattice orderings which are closely related to the total orderings on free groups.These orderings are natural in the sense that the imposed partial ordering on the idempotents coincides with the natural partial ordering. For this to happen in a lattice ordered inverse semigroup, the idempotents must form a distributive lattice. The method of construction of the lattice orderings on free inverse monoids can be applied to show that naturally lattice ordered inverse semigroups with a given distributive lattice E of idempotents can have arbitrary Green's relation structure. Analogous results hold for naturally ∧-semilatticed inverse semigroups. In this case, there is no restriction on the semilattice E of idempotents.We also show that every compatible lattice ordering on the free monogenic inverse monoid is of the type considered here. This permits us to prove that there are precisely eight distinct compatible lattice orderings on this semigroup. They belong to two families, each of which contains four members, of conjuguate lattice orderings.

We give an abstract description of the kernel of a proper primitive inverse congruence on a categorical semigroup. More specifically, we show that it is a *-reflexive, *-unitary, *-dense subsemigroup, and that on a given categorical semigroup there is a one–one correspondence between such subsemigroups and the proper primitive inverse congruences. Our results allow us to give a description of the minimum proper primitive inverse semigroup congruence on a strongly E*-dense categorical semigroup.

Margolis and Meakin use the Cayley graph of a group presentation to construct E-unitary inverse monoids [11]. This is the technique we refer to as graph expansion. In this paper we consider graph expansions of unipotent monoids, where a monoid is unipotent if it contains a unique idempotent. The monoids arising in this way are E-unitary and belong to the quasivariety of weakly left ample monoids. We give a number of examples of such monoids. We show that the least unipotent congruence on a weakly left ample monoid is given by the same formula as that for the least group congruence on an inverse monoid and we investigate the notion of proper for weakly left ample monoids. Using graph expansions we construct a functor Fe from the category U of unipotent monoids to the category PWLA of proper weakly left ample monoids. The functor Fe is an expansion in the sense of Birget and Rhodes [2]. If we equip proper weakly left ample monoids with an extra unary operation and denote the corresponding category by PWLA 0...

Much of the structure theory of inverse semigroups is based on constructing arbitrary inverse semigroups from groups and semilattices. It is known that E-unitary (or proper) inverse semigroups may be described as P-semigroups (McAlister), or inverse subsemigroups of semidirect products of a semilattice by a group (O'Carroll) or Cu-semigroups built over an inverse category acted upon by a group (Margolis and Pin). On the other hand, every inverse semigroup is known to have an E-unitary inverse cover (McAlister). The aim of this paper is to develop a similar theory for proper weakly left ample semigroups, a class with properties echoing those of inverse semigroups. We show how the structure of semigroups in this class is based on constructing semigroups from unipotent monoids and semilattices. The results corresponding to those of McAlister, O'Carroll and Margolis and Pin are obtained.

Munn's construction of a fundamental inverse semigroup T E from a semilattice E provides an important tool in the study of inverse semigroups. We present here a semigroup F E that plays for a class of E -semiadequate semigroups the role that T E plays for inverse semigroups. Every inverse semigroup with semilattice of idempotents E is E -semiadequate. There are however many interesting E -semiadequate semigroups that are not inverse, we consider various such examples arising from Schutzenberger products.

The relation [Rtilde] on a monoid S provides a natural generalisation of Green’s relation R. If every [Rtilde]-class of S contains an idempotentS is left semiabundant; if [Rtilde] is a left congruence then S satisfies(CL). Regular monoids, indeed left abundant monoids, are left semiabundant and satisfy(CL). However, the class of left semiabundant monoids is much larger, as we illustrate with a number of examples. This is the first of three related papers exploring the relationship between unipo-tent monoids and left semiabundancy. We consider the situations where the power enlargement or the Szendrei expansion of a monoid yields a left semiabundant monoid with(CL). Using the Szendrei expansion and the notion of the least unipotent monoid congruence σ on a monoid S, we construct functors is a left adjoint of F σ. Here U is the category of unipotent monoids and F is a category of left semiabundant monoids with properties echoing those of F-inverse monoids.

In this paper we describe idempotent pure regular extensions by inverse semigroups by means of quivers and actions of inverse semigroups, generalising the category and action of groups approach presented by Margolis and Pin for E -unitary inverse semigroups. By making use of these new tools, we can uniformly reprove O'Carroll's and Billhardt's characterisations of idempotent pure inverse extensions by inverse semigroups as L m -semigroups and as inverse subsemigroups of a λ-semidirect product of a semilattice by an inverse semigroup, respectively.

The structure of a categorical, E*-dense, E*-unitary E-semigroup S is elucidated in terms of a ‘B-quiver’, where B is a primitive inverse semigroup. In the case where S is strongly categorical, B is a Brandt semigroup. A covering theorem is also proved, to the effect that every categorical E*-dense E-semigroup has a cover which is a categorical, E*-dense, E*-unitary E-semigroup.

The result of Ballantine [1] to the effect that a singular matrix A is a product of k idempotent matrices if and only if the rank of I – A does not exceed k times the nullity of A is generalized to endomorphisms of a class of independence algebras.

The relation ℛ* is defined on a semigroup S by the rule that ℛ*b if and only if the elements a, b of S are related by the Green's relation ℛ in some oversemigroup of S. A semigroup S is an E-semigroup if its set E(S)of idempotents is a subsemilattice of S. A left adequate semigroup is an E-semigroup in which every ℛ*-class contains an idempotent. It is easy to see that, in fact, each ℛ*-class of a left adequate semigroup contains a unique idempotent [2]. We denote the idempotent in the ℛ*-class of a by a+.

The main subject of this paper is the description of the congruences on certain monoids of transformations on a finite chain X n with n elements. Namely, we consider the monoids OR n and POR n of all full, respectively partial, transformations on X n that preserve or reverse the orientation, as well as their respective submonoids OP n and POP n of all orientation-preserving elements. The inverse monoid PORI n of all injective elements of POR n is also considered.

In the terminology of Birget and Rhodes [3], an expansion is a functor F from the category of semigroups into some special category of semigroups such that there is a natural transformation η from F to the identity functor for which η s is surjective for every semigroup S . The three expansions introduced in [3] have proved to be of particular interest when applied to groups. In fact, as shown in [4], Ĝ (2) are isomorphic for any group G , is an E -unitary inverse monoid and the kernel of the homomorphism η G is the minimum group congruence on . Furthermore, if G is the free group on A , then the “cut-down to generators” which is a subsemigroup of is the free inverse semigroup on A . Essentially the same result was given by Margolis and Pin [12].

In this article, we first consider n × n upper-triangular matrices with entries in a given semiring k. Matrices of this form with invertible diagonal entries form a monoid B n (k). We show that B n (k) splits as a semidirect product of the monoid of unitriangular matrices U n (k) by the group of diagonal matrices. When the semiring is a field, B n (k) is actually a group and we recover a well-known result from the theory of groups and Lie algebras. Pursuing the analogy with the group case, we show that U n (k) is the ordered set product of n(n − 1)/2 commutative monoids (the root subgroups in the group case). Finally, we give two different presentations of the Schutzenberger product of n groups G 1,…, G n , given a monoid presentation 〈A i | R i 〉 of each group G i . We also obtain as a special case presentations for the monoid of all n × n unitriangular Boolean matrices.

If S is an inverse semigroup then E is a congruence on C(S). If S is a regular semigroup then each E -class of C(5) is a complete modular sublattice of C(5). (See [6]. ) In [5, Sec. 3] Petrich presents a few characterisations of E when S is an inverse semigroup. Here we prove that on a regular semigroup 5, the relation E restricted to RC(5) is a congruence. Also we extend Petrichfs results to the lattice RC(S) of a regular semigroup 5 and present a characterisation of the greatest element of each E-class. Characterisations of the least element of each E-class have been presented by Feigenbaum [I] and La Torte[4]. THE LATTICE RC(S) We use, whenever possible, the notation of Howie [3]. Recall first that a regular semigroup 5 is said to be R-u_~potent if its set of idempotents E(S) is a left reqular band, i.e. if E(5) satisfies the identity ere = el. In [7,1; 8, 1.1 ] it is shown that on a regular semigroup 5

SynopsisThe existence of a smallest inverse congruence on an orthodox semigroup is known. It is also known that a regular semigroup S is locally inverse and orthodox if and only if there exists a local isomorphism from S onto an inverse semigroup T.In this paper, we show the existence of a smallest R-unipotent congruence ρ on an orthodox semigroup S and give its expression in the case where S is also left quasinormal. Finally, we prove that a regular semigroup S is left quasinormal and orthodox if and only if there exists a local isomorphism from S onto an R-unipotent semigroup T.