Constellations and their relationship with categories

Constellations are partial algebras that are one-sided generalisations of categories. Indeed, we show that a category is exactly a constellation that also satisfies the left-right dual axioms. Constellations have previously appeared in the context of inductive constellations: the category of inductive constellations is known to be isomorphic to the category of left restriction semigroups. Here we consider constellations in full generality, giving many examples. We characterise those small constellations that are isomorphic to constellations of partial functions. We examine in detail the relationship between constellations and categories. In particular, we characterise those constellations that arise as (sub-)reducts of categories. We demonstrate that the notion of substructure can be captured within constellations but not within categories. We show that every constellation P gives rise to a category \({\mathcal{C}(P)}\), its canonical extension, in a simplest possible way, and that P is a quotient of \({\mathcal{C}(P)}\) in a natural sense. We also show that many of the most common concrete categories may be constructed from simpler quotient constellations using this construction. We characterise the canonical congruences \({\delta}\) on a given category \({K}\) (those for which \({K \cong \mathcal{C}(K/\delta))}\), and show that the category of constellations is equivalent to the category of \({\delta}\)-categories, that is, categories equipped with distinguished canonical congruence \({\delta}\).The main observation of this paper is that category theory as it applies to the familiar concrete categories of modern mathematics (which come equipped with natural notions of substructures and indeed are \({\delta}\)-categories) may be subsumed by constellation theory.     

Actions and partial actions of inductive constellations

Constellations were recently introduced by the authors as one-sided analogues of categories: a constellation is equipped with a partial multiplication for which ‘domains’ are defined but, in general, ‘ranges’ are not. Left restriction semigroups are the algebraic objects modelling semigroups of partial mappings, equipped with local identities in the domains of the mappings. Inductive constellations correspond to left restriction semigroups in a manner analogous to the correspondence between inverse semigroups and inductive groupoids.     

Algebras of Ehresmann semigroups and categories

E-Ehresmann semigroups are a commonly studied generalization of inverse semigroups. They are closely related to Ehresmann categories in the same way that inverse semigroups are related to inductive groupoids. We prove that under some finiteness condition, the semigroup algebra of an E-Ehresmann semigroup is isomorphic to the category algebra of the corresponding Ehresmann category. This generalizes a result of Steinberg who proved this isomorphism for inverse semigroups and inductive groupoids and a result of Guo and Chen who proved it for ample semigroups. We also characterize E-Ehresmann semigroups whose corresponding Ehresmann category is an EI-category and give some natural examples.     

Proper two-sided restriction semigroups and partial actions

Two-sided restriction semigroups and their handed versions arise from a number of sources. Attracting a deal of recent interest, they appear under a plethora of names in the literature. The class of left restriction semigroups essentially provides an axiomatisation of semigroups of partial mappings. It is known that this class admits proper covers, and that proper left restriction semigroups can be described by monoids acting on the left of semilattices. Any proper left restriction semigroup embeds into a semidirect product of a semilattice by a monoid, and moreover, this result is known in the wider context of left restriction categories. The dual results hold for right restriction semigroups.What can we say about two-sided restriction semigroups, hereafter referred to simply as restriction semigroups? Certainly, proper covers are known to exist. Here we consider whether proper restriction semigroups can be described in a natural way by monoids acting on both sides of a semilattice.It transpires that to obtain the full class of proper restriction semigroups, we must use partial actions of monoids, thus recovering results of Petrich and Reilly and of Lawson for inverse semigroups and ample semigroups, respectively. We also describe the class of proper restriction semigroups such that the partial actions can be mutually extendable to actions. Proper inverse and free restriction semigroups (which are proper) have this form, but we give examples of proper restriction semigroups which do not.     

On algebras of <Emphasis Type="Italic">P</Emphasis>-Ehresmann semigroups and their associate partial semigroups

P-Ehresmann semigroups are introduced by Jones as a common generalization of Ehresmann semigroups and regular \(*\)-semigroups. Ehresmann semigroups and their semigroup algebras are investigated by many authors in literature. In particular, Stein shows that under some finiteness condition, the semigroup algebra of an Ehresmann semigroup with a left (or right) restriction condition is isomorphic to the category algebra of the corresponding Ehresmann category. In this paper, we generalize this result to P-Ehresmann semigroups. More precisely, we show that for a left (or right) P-restriction locally Ehresmann P-Ehresmann semigroup \(\mathbf{S}\), if its projection set is principally finite, then we can give an algebra isomorphism between the semigroup algebra of \(\mathbf{S}\) and the partial semigroup algebra of the associate partial semigroup of \(\mathbf{S}\). Some interpretations and necessary examples are also provided to show why the above isomorphism dose not work for more general P-Ehresmann semigroups.     

Extending the Ehresmann-Schein-Nambooripad Theorem

We extend the ‘∨-premorphisms’ part of the Ehresmann-Schein-Nambooripad Theorem to the case of two-sided restriction semigroups and inductive categories, following on from a result of Lawson (J. Algebra 141:422–462, 1991) for the ‘morphisms’ part. However, it is so-called ‘∧-premorphisms’ which have proved useful in recent years in the study of partial actions. We therefore obtain an Ehresmann-Schein-Nambooripad-type theorem for (ordered) ∧-premorphisms in the case of two-sided restriction semigroups and inductive categories. As a corollary, we obtain such a theorem in the inverse case.     

Restriction semigroups and $$\lambda $$-Zappa-Szép products

The aim of this paper is to study \(\lambda \)-semidirect and \(\lambda \)-Zappa-Szép products of restriction semigroups. The former concept was introduced for inverse semigroups by Billhardt, and has been extended to some classes of left restriction semigroups. The latter was introduced, again in the inverse case, by Gilbert and Wazzan. We unify these concepts by considering what we name the scaffold of a Zappa-Szép product \(S\bowtie T\) where S and T are restriction. Under certain conditions this scaffold becomes a category. If one action is trivial, or if S is a semilattice and T a monoid, the scaffold may be ordered so that it becomes an inductive category. A standard technique, developed by Lawson and based on the Ehresmann-Schein-Nambooripad result for inverse semigroups, allows us to define a product on our category. We thus obtain restriction semigroups that are \(\lambda \)-semidirect products and \(\lambda \)-Zappa-Szép products, extending the work of Billhardt and of Gilbert and Wazzan. Finally, we explicate the internal structure of \(\lambda \)-semidirect products.     

On Ehresmann semigroups

We formulate an alternative approach to describing Ehresmann semigroups by means of left and right étale actions of a meet semilattice on a category. We also characterize the Ehresmann semigroups that arise as the set of all subsets of a finite category. As applications, we prove that every restriction semigroup can be nicely embedded into a restriction semigroup constructed from a category, and we describe when a restriction semigroup can be nicely embedded into an inverse semigroup.

     

Radical of weakly ordered semigroup algebras

We define the notion of weakly ordered semigroups. For this class of semigroups, we compute the radical of the semigroup algebras. This generalizes some results on left regular bands and on 0- Hecke algebras.     

On lattices of varieties of restriction semigroups

The left restriction semigroups have arisen in a number of contexts, one being as the abstract characterization of semigroups of partial maps, another as the ‘weakly left E-ample’ semigroups of the ‘York school’, and, more recently as a variety of unary semigroups defined by a set of simple identities. We initiate a study of the lattice of varieties of such semigroups and, in parallel, of their two-sided versions, the restriction semigroups. Although at the very bottom of the respective lattices the behaviour is akin to that of varieties of inverse semigroups, more interesting features are soon found in the minimal varieties that do not consist of semilattices of monoids, associated with certain ‘forbidden’ semigroups. There are two such in the one-sided case, three in the two-sided case. Also of interest in the one-sided case are the varieties consisting of unions of monoids, far indeed from any analogue for inverse semigroups. In a sequel, the author will show, in the two-sided case, that some rather surprising behavior is observed at the next ‘level’ of the lattice of varieties.     

Left and right negatively orderable semigroups and a one-sided version of Simon’s theorem

We study the classes \(\mathrm {LNO}\) and \(\mathrm {RNO}\) of left and right negatively orderable semigroups, arising as natural one-sided generalizations of negatively orderable semigroups (\(\mathrm {NO}\)). Negatively ordered monoids are well-known, for instance, from an equivalent formulation by Straubing and Thérien, of a celebrated theorem by I. Simon on piecewise testable languages. The main aim of this paper is to prove a one-sided version of Simon’s theorem for \(\mathrm {LNO}\) (\(\mathrm {RNO}\)). Analogues of some well-known results about negatively orderable semigroups are established in these cases. A characterisation of right negatively orderable semigroups as semigroups of certain type of decreasing mappings on partially ordered sets is obtained. We present sets of quasi-identities defining the quasivarieties \(\mathrm {LNO}\) and \(\mathrm {RNO}\) and show that these classes cannot be defined by finite sets of quasi-identities. We prove \(\mathrm {NO}=\mathrm {LNO}\cap \mathrm {RNO}\) and describe \(\mathrm {LNO}\vee \mathrm {RNO}\). As a one-sided version of Simon’s theorem, the pseudovariety generated by all finite semigroups in \(\mathrm {LNO}\) (\(\mathrm {RNO}\)) is determined and an Eilenberg-type correspondence between this pseudovariety and a variety of languages is established.     

Limit Automorphisms of the <Emphasis Type="Italic">C</Emphasis>*-Algebras Generated by Isometric Representations for Semigroups of Rationals

We consider inductive sequences of Toeplitz algebras whose connecting homomorphisms are defined by collections of primes. The inductive limits of these sequences are C*-algebras generated by representations for semigroups of rationals. We study the limit endomorphisms of these C*-algebras induced by morphisms between copies of the same inductive sequences of Toeplitz algebras. We establish necessary and sufficient conditions for these endomorphisms to be automorphisms of the algebras.     

Constellations with range and IS-categories

Constellations are asymmetric generalisations of categories. Although they are not required to possess a notion of range, many natural examples do. These include commonly occurring constellations related to concrete categories (since they model surjective morphisms), and also others arising from quite different sources, including from well-studied classes of semigroups. We show how constellations with a well-behaved range operation are nothing but ordered categories with restrictions. We characterise abstractly those categories that are canonical extensions of constellations with range, as so-called IS-categories. Such categories contain distinguished subcategories of insertions (which are monomorphisms) and surjections (in general different to the epimorphisms) such that each morphism admits a unique factorisation into a surjection followed by an insertion. Most familiar concrete categories are IS-categories, and we show how some of the well-known properties of these categories arise from the fact that they are IS-categories. For appropriate choices of morphisms in each, the category of IS-categories is shown to be equivalent to the category of constellations with range.     

The Generalized Prefix Expansion of a Weakly Left Ample Semigroup

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.     

关于Wrpp半群上的Lawson偏序(英文)

In this paper,we introduce Lawson partial order ≤ w on wrpp semigroups.After obtaining some properties of ≤ w,we determine when ≤ w is(left;right) compatible with the multiplication.These results extend and enrich the related results of Lawson and GuoLuo on abundant semigroups,of Guo-Shum on rpp semigroups and of Liu-Guo on wrpp semigroups.     

On Using the Join Operation to Define Classes of Algebras

We call a class of algebras a finitary prevariety if the class is closed under the formation of subalgebras and finitary direct products, and contains the one-element algebra. The join of two finitary prevarieties and a concept of a join-irreducible finitary prevariety may be introduced naturally. We develop techniques for proving that a finitary prevariety of semigroups is join-irreducible, and find many examples of join-irreducible finitary prevarieties of semigroups. For example, we prove that if a class of finite semigroups is defined by ω-identities and contains the class J, then it is a join-irreducible finitary prevariety.     

Rees matrix covers for a class of locally U-commutative semigroups

Given the importance of Morita theory of semigroups, we continue the study on the local structure of semigroups. Here we consider a class of nonregular semigroups, called locally U-commutative semigroups having U-local units, containing the classes of locally inverse semigroups, locally adequate semigroups, locally Ehresmann semigroups, and semigroups with local units having locally commuting idempotents. Our aim is to give a Rees matrix covering theorem for such semigroups with a partial McAlister sandwich bundle, and hence to put all the existing results into one context.     

On left regular and intra-regular ordered semigroups

We study the decomposition of left regular ordered semigroups into left regular components and the decomposition of intra-regular ordered semigroups into simple or intra-regular components, adding some additional information to the results considered in [KEHAYOPULU, N.: On left regular ordered semigroups, Math. Japon. 35 (1990), 1057–1060] and [KEHAYOPULU, N.: On intra-regular ordered semigroups, Semigroup Forum 46 (1993), 271–278]. We prove that an ordered semigroup S is left regular if and only if it is a semilattice (or a complete semilattice) of left regular semigroups, equivalently, it is a union of left regular subsemigroups of S. Moreover, S is left regular if and only if it is a union of pairwise disjoint left regular subsemigroups of S. The right analog also holds. The same result is true if we replace the words “left regular” by “intraregular”. Moreover, an ordered semigroup is intra-regular if and only if it is a semilattice (or a complete semilattice) of simple semigroups. On the other hand, if an ordered semigroup is a semilattice (or a complete semilattice) of left simple semigroups, then it is left regular, but the converse statement does not hold in general. Illustrative examples are given.     

Quiver approach to some ordered semigroup algebras

The aim of this paper is to research the relation among generalized path algebras, pseudo-admissible ordered semigroup algebras and path algebras over algebras. First, the Gabriel theorem for contract pseudo-admissible ordered semigroup algebras is given. Second, a family of pseudo-admissible ordered semigroups is mixed together via the method of generalized path algebras to construct a new pseudo-admissible ordered semigroup as the extended version. Finally, we characterize the path algebra over an algebra in two ways through normal and non-normal generalized path algebras, respectively, over a field.