Dual Space of a Lattice as the Completion of a Pervin Space

Pin, Jean-Éric

doi:10.1007/978-3-319-57418-9_2

Cited by 7 publications

(24 citation statements)

References 18 publications

(17 reference statements)

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“…Actually, more can be said: every transitive and totally bounded quasi-uniformity on a set X is of the form E F for some family F ⊆ P(X). To the best of our knowledge, this result is due to Gehrke, Grigorieff, and Pin in unpublished work (see also [18]), and at the present date, a proof can be found in the Goubault-Larrecq's blog [8]. Another interesting aspect of Pervin spaces is that the bounded sublattice of P(X) generated by some family F can be recovered from the quasi-uniform space (X, E F ): it consists of the subsets U ⊆ P(X) such that E U ∈ E F (see [7,Theorem 5.1] for a proof).…”

Section: Introductionsupporting

confidence: 59%

“…Note that, unlike what happens in the point-set framework, where we have an equivalence between Pervin spaces and transitive and totally bounded quasi-uniform spaces (see [18]), the categories of Frith frames and of transitive and totally bounded quasi-uniform frames are not equivalent. This is because, in general, the dense extremal epimorphism γ (K, E) : (C S L, E S ) ։ (K, E) from Proposition 5.11 is not an isomorphism.…”

Section: Frith Frames As Quasi-uniform Framesmentioning

confidence: 85%

“…However, since quasi-uniform spaces are not the subject of this paper, but rather its pointfree counterpart, we will not provide further details on this matter. We refer the reader to [13,4,18] for further reading.…”

Section: Pervin Spacesmentioning

confidence: 99%

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A pointfree theory of Pervin spaces

Borlido¹,

Suarez²

2022

Preprint

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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.

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Section: Introductionsupporting

confidence: 59%

Section: Frith Frames As Quasi-uniform Framesmentioning

confidence: 85%

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A pointfree theory of Pervin spaces

Borlido¹,

Suarez²

2022

Preprint

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“…6 A key part of our computation is the description of the Stone space of the set of all reverse definite languages over a fixed alphabet. This space is actually described in [32] though for completely different reasons from ours.…”

Section: Applications and Examplesmentioning

confidence: 75%

The Booleanization of an inverse semigroup

Lawson

2019

Semigroup Forum

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We prove that the forgetful functor from the category of Boolean inverse semigroups to the category of inverse semigroups with zero has a left adjoint. This left adjoint is what we term the 'Booleanization'. We establish the exact theoretical connection between the Booleanization of an inverse semigroup and Paterson's universal groupoid of the inverse semigroup and we explicitly compute the concrete Booleanization of the polycyclic inverse monoid Pn and demonstrate its affiliation with the Cuntz-Toeplitz algebra.

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“…The inequality theory for languages was first introduced in [14] and later used in [15,16,22]. It is based on the following definitions.…”

Section: Inequalities On Languagesmentioning

confidence: 99%

Inequalities for One-Step Products

Branco

2018

Developments in Language Theory

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Let a be a letter of an alphabet A. Given a lattice of languages L, we describe the set of ultrafilter inequalities satisfied by the lattice La generated by the languages of the form L or LaA * , where L is a language of L. We also describe the ultrafilter inequalities satisfied by the lattice L1 generated by the lattices La, for a ∈ A. When L is a lattice of regular languages, we first describe the profinite inequalities satisfied by La and L1 and then provide a small basis of inequalities defining L1 when L is a Boolean algebra of regular languages closed under quotient.

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Dual Space of a Lattice as the Completion of a Pervin Space

Abstract: We assume the reader is familiar with basic topology on the one hand and finite automata theory on the other hand. No proofs are given in this extended abstract.

Cited by 7 publications

References 18 publications

A pointfree theory of Pervin spaces

A pointfree theory of Pervin spaces

The Booleanization of an inverse semigroup

Inequalities for One-Step Products

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