scholarly journals A complete Heyting algebra whose Scott space is non-sober

2021 ◽  
Vol 252 (3) ◽  
pp. 315-323 ◽  
Author(s):  
Xiaoquan Xu ◽  
Xiaoyong Xi ◽  
Dongsheng Zhao
Keyword(s):  
Heyting Algebra

scholarly journals When do L-fuzzy ideals of a ring generate a distributive lattice?

2016 ◽  
Vol 14 (1) ◽  
pp. 531-542
Author(s):  
Ninghua Gao ◽  
Qingguo Li ◽  
Zhaowen Li
Keyword(s):  
Distributive Lattice ◽  
Heyting Algebra ◽  
Lattice Structures ◽  
Boolean Ring ◽  
Fuzzy Subset ◽  
The Family ◽  
Essential Properties ◽  
Fuzzy Ideal

AbstractThe notion of L-fuzzy extended ideals is introduced in a Boolean ring, and their essential properties are investigated. We also build the relation between an L-fuzzy ideal and the class of its L-fuzzy extended ideals. By defining an operator “⇝” between two arbitrary L-fuzzy ideals in terms of L-fuzzy extended ideals, the result that “the family of all L-fuzzy ideals in a Boolean ring is a complete Heyting algebra” is immediately obtained. Furthermore, the lattice structures of L-fuzzy extended ideals of an L-fuzzy ideal, L-fuzzy extended ideals relative to an L-fuzzy subset, L-fuzzy stable ideals relative to an L-fuzzy subset and their connections are studied in this paper.

scholarly journals On an interpretation of second order quantification in first order intuitionistic propositional logic

1992 ◽  
Vol 57 (1) ◽  
pp. 33-52 ◽  
Author(s):  
Andrew M. Pitts
Keyword(s):  
Propositional Logic ◽  
Heyting Algebra ◽  
Propositional Calculus ◽  
Interpolation Theorem ◽  
Second Order ◽  
Heyting Algebras ◽  
Intuitionistic Propositional Logic ◽  
First Order ◽  
Intuitionistic Propositional Calculus ◽  
Algebraic Counterpart

AbstractWe prove the following surprising property of Heyting's intuitionistic propositional calculus, IpC. Consider the collection of formulas, ϕ, built up from propositional variables (p, q, r, …) and falsity (⊥) using conjunction (∧), disjunction (∨) and implication (→). Write ⊢ϕ to indicate that such a formula is intuitionistically valid. We show that for each variable p and formula ϕ there exists a formula Apϕ (effectively computable from ϕ), containing only variables not equal to p which occur in ϕ, and such that for all formulas ψ not involving p, ⊢ψ → Apϕ if and only if ⊢ψ → ϕ. Consequently quantification over propositional variables can be modelled in IpC, and there is an interpretation of the second order propositional calculus, IpC2, in IpC which restricts to the identity on first order propositions.An immediate corollary is the strengthening of the usual interpolation theorem for IpC to the statement that there are least and greatest interpolant formulas for any given pair of formulas. The result also has a number of interesting consequences for the algebraic counterpart of IpC, the theory of Heyting algebras. In particular we show that a model of IpC2 can be constructed whose algebra of truth-values is equal to any given Heyting algebra.

The subalgebra lattice of a Heyting algebra

1987 ◽  
Vol 37 (1) ◽  
pp. 34-41 ◽  
Author(s):  
L. Vrancken-Mawet ◽  
Georges Hansoul
Keyword(s):  
Heyting Algebra ◽  
Subalgebra Lattice

AUTOMATED COMPOSITIONAL REASONING OF INTUITIONISTICALLY CLOSED REGULAR PROPERTIES

2009 ◽  
Vol 20 (04) ◽  
pp. 747-762
Author(s):  
YIH-KUEN TSAY ◽  
BOW-YAW WANG
Keyword(s):  
Learning Algorithms ◽  
Heyting Algebra ◽  
Regular Languages ◽  
Safety Properties ◽  
Compositional Reasoning ◽  
Proof Rules

Analysis of infinitary safety properties with automated compositional reasoning through learning is discussed in the paper. We consider the class of intuitionistically closed regular languages and show that it forms a Heyting algebra and is finitely approximatable. Subsequently, compositional proof rules can be verified automatically and learning algorithms for finitary regular languages suffice. We also establish an axiom to deduce circular compositional proof rules for the infinitary languages.

Construction of a Monadic Heyting Algebra in a Logos

2016 ◽  
Vol 218 (6) ◽  
pp. 788-793
Author(s):  
A. Klimiashvili
Keyword(s):  
Heyting Algebra

scholarly journals A Heyting Algebra on Dyck Paths of Type A and B

Order ◽  
2016 ◽  
Vol 34 (2) ◽  
pp. 327-348 ◽  
Author(s):  
Henri Mühle
Keyword(s):  
Heyting Algebra ◽  
Type A ◽  
Dyck Paths

scholarly journals (T,N) and Residual Fuzzy Co-Implication in Dual Heyting Algebra with Applications

2016 ◽  
Author(s):  
Iqbal H. Jebril
Keyword(s):  
Heyting Algebra ◽  
Fuzzy Implications ◽  
Fuzzy Implication ◽  
New Concepts

Recently, many authors have been interested to introduce fuzzy implications over t-norms and t-conorms. In this paper, we introduce (S,N) and residuum fuzzy implication for Dubois t-norm and Hamacher's t-norm. Also, new concepts so-called (T,N) and residual fuzzy co-implication in dual Heyting Algebra are investigated. Some examples as well as application are discussed as well.

scholarly journals Boolean algebras in algebraic topology

2007 ◽  
pp. 135-139
Author(s):  
Zoran Petrovic
Keyword(s):  
Algebraic Topology ◽  
Heyting Algebra ◽  
Boolean Algebras ◽  
Main Topic

The appearance of the complete Heyting algebra in the realm of Algebraic Topology is the main topic of the paper.

scholarly journals Soft Set Theory Applied to Hoops

2020 ◽  
Vol 28 (1) ◽  
pp. 61-79
Author(s):  
R. A. Borzooei ◽  
E. Babaei ◽  
Y. B. Jun ◽  
M. Aaly Kologani ◽  
M. Mohseni Takallo
Keyword(s):  
Set Theory ◽  
Congruence Relation ◽  
Heyting Algebra ◽  
Soft Set ◽  
The Family ◽  
Different Types

AbstractIn this paper, we introduced the concept of a soft hoop and we investigated some of their properties. Then, we established different types of intersections and unions of the family of soft hoops. We defined two operations ⊙ and → on the set of all soft hoops and we proved that with these operations, it is a hoop and also is a Heyting algebra. Finally we introduced a congruence relation on the set of all soft hoops and we investigated the quotient of it.

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