Bi-Heyting algebras, toposes and modalities

Reyes, Gonzalo E.; Zolfaghari, Houman

doi:10.1007/bf00357841

Cited by 53 publications

(28 citation statements)

References 7 publications

Supporting

Mentioning

Contrasting

Unclassified

Order By: Relevance

“…In [8], it was shown that any complete bi-Heyting algebra comes equipped with a canonical modal structure that is obtained through the iteration of the Heyting and co-Heyting negations of the algebra. We will now give a brief overview of this modal structure.…”

Section: Modal Operators In Tqtmentioning

confidence: 99%

See 1 more Smart Citation

Modality and Contextuality in Topos Quantum Theory

Eva

2016

Stud Logica

View full text Add to dashboard Cite

Abstract.Topos quantum theory (TQT) represents a whole new approach to the formalization of non-relativistic quantum theory. It is well known that TQT replaces the orthomodular quantum logic of the traditional Hilbert space formalism with a new intuitionistic logic that arises naturally from the topos theoretic structure of the theory. However, it is less well known that TQT also has a dual logical structure that is paraconsistent. In this paper, we investigate the relationship between these two logical structures and study the implications of this relationship for the definition of modal operators in TQT.

show abstract

Section: Modal Operators In Tqtmentioning

confidence: 99%

“…(Reyes and Zolfaghari [8]) Let a ∈ B. Then a is the greatest complemented x ∈ B such that x ≤ a and ♦a is the least complemented x ∈ B such that x ≥ a.…”

Section: Modal Operators In Tqtmentioning

confidence: 99%

Modality and Contextuality in Topos Quantum Theory

Eva

2016

Stud Logica

View full text Add to dashboard Cite

show abstract

“…This sublattice is not a Boolean algebra as it is not closed under complementation. It is a bi-Heyting algebra for which the special case of graphs rather than hypergraphs has been discussed by Lawvere [18] and Reyes and Zolfaghari [19]. Figure 2 shows a subgraph K, indicated by the unbroken edges and the nodes as solid discs, together with its pseudocomplement and dual pseudocomplement.…”

Section: Definition 10mentioning

confidence: 99%

Symmetric Heyting relation algebras with applications to hypergraphs

Stell

2015

Journal of Logical and Algebraic Methods in Programming

View full text Add to dashboard Cite

Article:Stell, JG (2014) Symmetric Heyting relation algebras with applications to hypergraphs. Journal of Logical and Algebraic Methods in Programming, 84. pp. 440-455. ISSN 2352-2208 https://doi.org/10.1016/j.jlamp.2014.12.001 eprints@whiterose.ac.uk https://eprints.whiterose.ac.uk/ Reuse Unless indicated otherwise, fulltext items are protected by copyright with all rights reserved. The copyright exception in section 29 of the Copyright, Designs and Patents Act 1988 allows the making of a single copy solely for the purpose of non-commercial research or private study within the limits of fair dealing. The publisher or other rights-holder may allow further reproduction and re-use of this version -refer to the White Rose Research Online record for this item. Where records identify the publisher as the copyright holder, users can verify any specific terms of use on the publisher's website. TakedownIf you consider content in White Rose Research Online to be in breach of UK law, please notify us by emailing eprints@whiterose.ac.uk including the URL of the record and the reason for the withdrawal request. AbstractA relation on a hypergraph is a binary relation on the set consisting of all the nodes and the edges, and which satisfies a constraint involving the incidence structure of the hypergraph. These relations correspond to join preserving mappings on the lattice of sub-hypergraphs. This paper introduces a generalization of a relation algebra in which the Boolean algebra part is replaced by a Heyting algebra that supports an order-reversing involution. A general construction for these symmetric Heyting relation algebras is given which includes as a special case the algebra of relations on a hypergraph. A particular feature of symmetric Heyting relation algebras is that instead of an involutory converse operation they possess both a left converse and a right converse which form an adjoint pair of operations. Properties of the converses are established and used to derive a generalization of the well-known connection between converse, complement, erosion and dilation in mathematical morphology. This provides part of the foundation necessary to develop mathematical morphology on hypergraphs based on relations on hypergraphs.

show abstract

“…This bi-Heyting structure has been described in [Law86,RZ96], but we review it here, in the more general setting of hypergraphs, as it provides motivation for later sections.…”

Section: The Algebra Of Graphs and Hypergraphsmentioning

confidence: 99%

Formal Concept Analysis over Graphs and Hypergraphs

Stell

2014

Lecture Notes in Computer Science

View full text Add to dashboard Cite

Bi-Heyting algebras, toposes and modalities

Cited by 53 publications

References 7 publications

Modality and Contextuality in Topos Quantum Theory

Modality and Contextuality in Topos Quantum Theory

Symmetric Heyting relation algebras with applications to hypergraphs

Formal Concept Analysis over Graphs and Hypergraphs

Contact Info

Product

Resources

About