arXiv subjects

Célia Borlido

Publications and source records attributed to Célia Borlido.

13 recordsLinked to original sources

arXiv2025

Difference-restriction algebras with operators

We exhibit an adjunction between a category of abstract algebras of partial functions that we call difference-restriction algebras and a category of Hausdorff étale spaces. Difference-restriction algebras are those algebras isomorphic to a collection of partial functions closed under relative complement and domain restriction. Our adjunction generalises the adjunction between the category of generalised Boolean algebras and the category of Hausdorff spaces. We define the finitary compatible completion of a difference-restriction algebra and show that the monad induced by our adjunction yields the finitary compatible completion of any difference-restriction algebra. As a corollary, the adjunction restricts to a duality between the finitarily compatibly complete difference-restriction algebras and the locally compact zero-dimensional Hausdorff étale spaces, generalising the duality between generalised Boolean algebras and locally compact zero-dimensional Hausdorff spaces. We then extend these adjunction, duality, and completion results to difference-restriction algebras equipped with arbitrary additional compatibility preserving operators.

math.LO↗

arXiv2024

Filtral pretoposes and compact Hausdorff locales

The category of compact Hausdorff locales is a pretopos which is filtral, meaning that every object is covered by one whose subobject lattice is isomorphic to the lattice of filters of complemented elements. We show that any filtral pretopos satisfying some mild additional conditions can be embedded into the category of compact Hausdorff locales. This result is valid in the internal logic of any topos. Assuming the principle of weak excluded middle and the existence of copowers of the terminal object in the pretopos, the image of the embedding contains all spatial compact Hausdorff locales. The notion of filtrality was introduced by V. Marra and L. Reggio (Theory Appl. Categ., 2020) to characterise the category of compact Hausdorff spaces within the class of pretoposes. Our results can be regarded as a constructive extension of the aforementioned characterisation, avoiding reference to points. If the ambient logic is classical, i.e. it satisfies excluded middle, and the prime ideal theorem for Boolean algebras holds, we obtain as a corollary the characterisation of compact Hausdorff spaces in op. cit.

math.CT↗

arXiv2024

On semidirect products of quantale enriched monoids

We consider monoids equipped with a compatible quantale valued relation, to which we call quantale enriched monoids, and study semidirect products of such structures. It is well-known that semidirect products of monoids are closely related to Schreier split extensions which, in the setting of monoids, play the role of split extensions of groups. We will thus introduce certain split extensions of quantale enriched monoids, which generalize the classical Schreier split extensions of monoids, and investigate their connections with semidirect products. We then restrict our study to a class of quantale enriched monoids whose behavior mimics the fact that the preorder on a preordered group is completely determined by its cone of positive elements. Finally, we instantiate our results for preordered monoids and compare them with existing literature.

math.GR↗

arXiv2024

Pervin spaces and Frith frames: bitopological aspects and completion

A Pervin space is a set equipped with a bounded sublattice of its powerset, while its pointfree version, called Frith frame, consists of a frame equipped with a generating bounded sublattice. It is known that the dual adjunction between topological spaces and frames extends to a dual adjunction between Pervin spaces and Frith frames, and that the latter may be seen as representatives of certain quasi-uniform structures. As such, they have an underlying bitopological structure and inherit a natural notion of completion. In this paper we start by exploring the bitopological nature of Pervin spaces and of Frith frames, proving some categorical equivalences involving zero-dimensional structures. We then provide a conceptual proof of a duality between the categories of $T_0$ complete Pervin spaces and of complete Frith frames. This enables us to interpret several Stone-type dualities as a restriction of the dual adjunction between Pervin spaces and Frith frames along full subcategory embeddings. Finally, we provide analogues of Banaschewski and Pultr's characterizations of sober and $T_D$ topological spaces in the setting of Pervin spaces and of Frith frames, highlighting the parallelism between the two notions.

math.GN↗

arXiv2023

A pointfree theory of Pervin spaces

We lay down the foundations for a pointfree theory of Pervin spaces. A Pervin space is a set equipped with a bounded sublattice of its powerset, and it is known that these objects characterize those quasi-uniform spaces that are transitive and totally bounded. The pointfree notion of a Pervin space, which we call Frith frame, consists of a frame equipped with a generating bounded sublattice. In this paper we introduce and study the category of Frith frames and show that the classical dual adjunction between topological spaces and frames extends to a dual adjunction between Pervin spaces and Frith frames. Unlike what happens for Pervin spaces, we do not have an equivalence between the categories of transitive and totally bounded quasi-uniform frames and of Frith frames, but we show that the latter is a full coreflective subcategory of the former. We also explore the notion of completeness of Frith frames inherited from quasi-uniform frames, providing a characterization of those Frith frames that are complete and a description of the completion of an arbitrary Frith frame.

math.GN↗

arXiv2023

Substitution Principle and semidirect products

In the classical theory of regular languages the concept of recognition by profinite monoids is an important tool. Beyond regularity, Boolean spaces with internal monoids (BiMs) were recently proposed as a generalization. On the other hand, fragments of logic defining regular languages can be studied inductively via the so-called "Substitution Principle". In this paper we make the logical underpinnings of this principle explicit and extend it to arbitrary languages using Stone duality. Subsequently we show how it can be used to obtain topo-algebraic recognizers for classes of languages defined by a wide class of first-order logic fragments. This naturally leads to a notion of semidirect product of BiMs extending the classical such construction for profinite monoids. Our main result is a generalization of Almeida and Weil's Decomposition Theorem for semidirect products from the profinite setting to that of BiMs. This is a crucial step in a program to extend the profinite methods of regular language theory to the setting of complexity theory.

cs.LO↗

arXiv2022

Difference-restriction algebras of partial functions: axiomatisations and representations

We investigate the representation and complete representation classes for algebras of partial functions with the signature of relative complement and domain restriction. We provide and prove the correctness of a finite equational axiomatisation for the class of algebras representable by partial functions. As a corollary, the same equations axiomatise the algebras representable as injective partial functions. For complete representations, we show that a representation is meet complete if and only if it is join complete. Then we show that the class of completely representable algebras is precisely the class of atomic and representable algebras. As a corollary, the same properties axiomatise the class of algebras completely representable by injective partial functions. The universal-existential-universal axiomatisation this yields for these complete representation classes is the simplest possible, in the sense that no existential-universal-existential axiomatisation exists.

math.LO↗

arXiv2022

Difference-restriction algebras of partial functions with operators: discrete duality and completion

We exhibit an adjunction between a category of abstract algebras of partial functions and a category of set quotients. The algebras are those atomic algebras representable as a collection of partial functions closed under relative complement and domain restriction; the morphisms are the complete homomorphisms. This generalises the discrete adjunction between the atomic Boolean algebras and the category of sets. We define the compatible completion of a representable algebra, and show that the monad induced by our adjunction yields the compatible completion of any atomic representable algebra. As a corollary, the adjunction restricts to a duality on the compatibly complete atomic representable algebras, generalising the discrete duality between complete atomic Boolean algebras and sets. We then extend these adjunction, duality, and completion results to representable algebras equipped with arbitrary additional completely additive and compatibility preserving operators.

math.LO↗

arXiv2019

A note on powers of Boolean spaces with internal semigroups

Boolean spaces with internal semigroups generalize profinite semigroups and are pertinent for the recognition of not-necessarily regular languages. Via recognition, the study of existential quantification in logic on words amounts to the study of certain spans of Boolean spaces with internal semigroups. In turn, these can be understood as the superposition of a span of Boolean spaces and a span of semigroups. In this note, we first study these separately. More precisely, we identify the conditions under which each of these spans gives rise to a morphism into the respective power or Vietoris construction of the corresponding structure. Combining these characterizations, we obtain such a characterization for spans of Boolean spaces with internal semigroups which we use to describe the topo-algebraic counterpart of monadic second-order existential quantification. This is closely related to a part of the earlier work on existential quantification in first-order logic on words by Gehrke, Petri\c san and Reggio. The observation that certain morphisms lift contravariantly to the appropriate power structures makes our analysis very simple.

math.GN↗

arXiv2018

Difference hierarchies and duality with an application to formal languages

The notion of a difference hierarchy, first introduced by Hausdorff, plays an important role in many areas of mathematics, logic and theoretical computer science such as descriptive set theory, complexity theory, and the theory of regular languages and automata. From a lattice theoretic point of view, the difference hierarchy over a bounded distributive lattice stratifies the Boolean algebra generated by it according to the minimum length of difference chains required to describe the Boolean elements. While each Boolean element is given by a finite difference chain, there is no canonical such writing in general. We show that, relative to the filter completion, or equivalently, the lattice of closed upsets of the dual Priestley space, each Boolean element over the lattice has a canonical minimum length decomposition into a Hausdorff difference. As a corollary each Boolean element over a (co-)Heyting algebra has a canonical difference chain. With a further generalization of this result involving a directed family of adjunctions with meet-semilattices, we give an elementary proof of the fact that a regular language is given by a Boolean combination of purely universal sentences using arbitrary numerical predicates if and only if it is given by a Boolean combination of purely universal sentences using only regular numerical predicates.

math.GN↗

arXiv2015

Some reducibility properties for pseudovarieties of the form DRH

Let H be a pseudovariety of groups and DRH be the pseudovariety containing all finite semigroups whose regular R-classes belong to H. We study the relationship between reducibility of H and of DRH with respect to several particular classes of systems of equations. The classes of systems considered (of pointlike, idempotent pointlike and graph equations) are known to play a role in decidability questions concerning pseudovarieties of the forms V * W, V join W, and V malcev W.

math.GR↗

arXiv2015

The kappa-word problem over DRH

Let H be a pseudovariety of groups in which the kappa-word problem is decidable. Here, kappa denotes the canonical implicit signature, which consists of the multiplication and the (omega-1)-power. We prove that the kappa-word problem is also decidable over DRH, the pseudovariety of all finite semigroups whose regular R-classes lie in H. Further, we present a canonical form for elements in the free kappa-semigroup over DRH, based on the knowledge of a canonical form for elements in the free kappa-semigroup over H. This extends work of Almeida and Zeitoun on the pseudovariety of all finite R-trivial semigroups.

math.GR↗

arXiv2015

Complete kappa-reducibility of pseudovarieties of the form DRH

We denote by kappa the implicit signature that contains the multiplication and the (omega-1)-power. It is proved that for any completely kappa-reducible pseudovariety of groups H, the pseudovariety DRH of all finite semigroups whose regular R-classes are groups in H is completely kappa-reducible as well. The converse also holds. The tools used by Almeida, Costa, and Zeitoun for proving that the pseudovariety of all finite R-trivial monoids is completely kappa-reducible are adapted for the general setting of a pseudovariety of the form DRH.

math.GR↗