A categorical equivalence for bounded distributive quasi lattices satisfying: x ∨ 0 = 0 ⇒ x = 0

Hector Freytes; Antonio Ledda

doi:10.2478/s12175-014-0262-2

A categorical equivalence for bounded distributive quasi lattices satisfying: x ∨ 0 = 0 ⇒ x = 0

Mathematica Slovaca ◽

10.2478/s12175-014-0262-2 ◽

2014 ◽

Vol 64 (5) ◽

Author(s):

Hector Freytes ◽

Antonio Ledda

Keyword(s):

Categorical Equivalence ◽

Priestley Spaces

AbstractIn this work, we investigate a categorical equivalence between the class of bounded distributive quasi lattices that satisfy the quasiequation x∨0 = 0 =⇒ x = 0, and a category whose objects are sheaves over Priestley spaces.

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Bitopological duality for distributive lattices and Heyting algebras

Mathematical Structures in Computer Science ◽

10.1017/s0960129509990302 ◽

2010 ◽

Vol 20 (3) ◽

pp. 359-393 ◽

Cited By ~ 17

Author(s):

GURAM BEZHANISHVILI ◽

NICK BEZHANISHVILI ◽

DAVID GABELAIA ◽

ALEXANDER KURZ

Keyword(s):

Boolean Algebras ◽

Distributive Lattices ◽

Heyting Algebras ◽

Priestley Spaces ◽

Spectral Spaces ◽

Algebraic Concepts ◽

Stone Spaces

We introduce pairwise Stone spaces as a bitopological generalisation of Stone spaces – the duals of Boolean algebras – and show that they are exactly the bitopological duals of bounded distributive lattices. The category PStone of pairwise Stone spaces is isomorphic to the category Spec of spectral spaces and to the category Pries of Priestley spaces. In fact, the isomorphism of Spec and Pries is most naturally seen through PStone by first establishing that Pries is isomorphic to PStone, and then showing that PStone is isomorphic to Spec. We provide the bitopological and spectral descriptions of many algebraic concepts important in the study of distributive lattices. We also give new bitopological and spectral dualities for Heyting algebras, thereby providing two new alternatives to Esakia's duality.

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Erratum: “A categorical equivalence between generalized holonomy maps on a connected manifold and principal connections on bundles over that manifold” [J. Math. Phys. 57, 102902 (2016)]

Journal of Mathematical Physics ◽

10.1063/1.5019748 ◽

2018 ◽

Vol 59 (2) ◽

pp. 029901

Author(s):

Sarita Rosenstock ◽

James Owen Weatherall

Keyword(s):

Categorical Equivalence ◽

Connected Manifold ◽

Math Phys

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A categorical equivalence between semi-Heyting algebras and centered semi-Nelson algebras

Logic Journal of IGPL ◽

10.1093/jigpal/jzy006 ◽

2018 ◽

Vol 26 (4) ◽

pp. 408-428 ◽

Cited By ~ 1

Author(s):

Juan Manuel Cornejo ◽

Hernán Javier San Martín

Keyword(s):

Heyting Algebras ◽

Categorical Equivalence ◽

Nelson Algebras

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MORITA EQUIVALENCE

The Review of Symbolic Logic ◽

10.1017/s1755020316000186 ◽

2016 ◽

Vol 9 (3) ◽

pp. 556-582 ◽

Cited By ~ 21

Author(s):

THOMAS WILLIAM BARRETT ◽

HANS HALVORSON

Keyword(s):

Morita Equivalence ◽

Categorical Equivalence ◽

Theoretical Equivalence

AbstractLogicians and philosophers of science have proposed various formal criteria for theoretical equivalence. In this paper, we examine two such proposals: definitional equivalence and categorical equivalence. In order to show precisely how these two well-known criteria are related to one another, we investigate an intermediate criterion called Morita equivalence.

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Proof of a Conjecture of S. Mac Lane

BRICS Report Series ◽

10.7146/brics.v3i61.18773 ◽

1996 ◽

Vol 3 (61) ◽

Author(s):

Sergei Soloviev

Keyword(s):

Linear Logic ◽

Sufficient Conditions ◽

Vector Spaces ◽

Categorical Equivalence ◽

Closed Category ◽

Symmetric Monoidal Closed Category ◽

Monoidal Closed Category ◽

Coherence Theorem ◽

Mac Lane

Some sufficient conditions on a Symmetric Monoidal Closed category K are obtained such that a diagram in a free SMC category generated by the set A of atoms commutes if and only if all its interpretations in K are commutative. In particular, the category of vector spaces on any field satisfies these conditions (only this case was considered in the original Mac Lane conjecture). Instead of diagrams, pairs of derivations in Intuitionistic Multiplicative Linear logic can be considered (together with categorical equivalence). Two derivations of the same sequent are equivalent if and only if all their interpretations in K are equal. In fact, the assignment of values (objects of K) to atoms is defined constructively for each pair of derivations. Taking into account a mistake in R. Voreadou's proof of the "abstract coherence theorem" found by the author, it was necessary to modify her description of the class of non-commutative diagrams in SMC categories; our proof of S. Mac Lane conjecture proves also the correctness of the modified description.

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