Heyting Algebras and Closure Algebras

2019 ◽  
pp. 15-40
Author(s):  
Leo Esakia
Keyword(s):  
Heyting Algebras ◽  
Closure Algebras

Complexity of interpolation and related problems in positive calculi

2002 ◽  
Vol 67 (1) ◽  
pp. 397-408 ◽  
Author(s):  
Larisa Maksimova
Keyword(s):  
Modal Logic ◽  
Heyting Algebras ◽  
Logical System ◽  
Complexity Bounds ◽  
Np Completeness ◽  
Interpolation Problems ◽  
Finite Set ◽  
Modal Logic S4 ◽  
Closure Algebras ◽  
Logical Calculi

AbstractWe consider the problem of recognizing important properties of logical calculi and find complexity bounds for some decidable properties. For a given logical system L, a property P of logical calculi is called decidable over L if there is an algorithm which for any finite set Ax of new axiom schemes decides whether the calculus L + Ax has the property P or not. In [11] the complexity of tabularity, pre-tabularity. and interpolation problems over the intuitionistic logic Int and over modal logic S4 was studied, also we found the complexity of amalgamation problems in varieties of Heyting algebras and closure algebras.In the present paper we deal with positive calculi. We prove NP-completeness of tabularity, DP-hardness of pretabularity and PSPACE-completeness of interpolation problem over Int+. In addition to above-mentioned properties, we consider Beth's definability properties. Also we improve some complexity bounds for properties of superintuitionistic calculi.

Topo-canonical completions of closure algebras and Heyting algebras

2007 ◽  
Vol 58 (1) ◽  
pp. 1-34 ◽  
Author(s):  
Guram Bezhanishvili ◽  
Ray Mines ◽  
Patrick J. Morandi
Keyword(s):  
Heyting Algebras ◽  
Closure Algebras

scholarly journals Residuated Structures and Orthomodular Lattices

Studia Logica ◽  
2021 ◽  
Author(s):  
D. Fazio ◽  
A. Ledda ◽  
F. Paoli
Keyword(s):  
Algebraic Logic ◽  
Residuated Lattices ◽  
Quantum Structures ◽  
Heyting Algebras ◽  
Quantum Algebras ◽  
Orthomodular Lattices ◽  
Lattice Of Subvarieties ◽  
Residuated Structures ◽  
Common Framework ◽  
Mv Algebras

AbstractThe variety of (pointed) residuated lattices includes a vast proportion of the classes of algebras that are relevant for algebraic logic, e.g., $$\ell $$ ℓ -groups, Heyting algebras, MV-algebras, or De Morgan monoids. Among the outliers, one counts orthomodular lattices and other varieties of quantum algebras. We suggest a common framework—pointed left-residuated $$\ell $$ ℓ -groupoids—where residuated structures and quantum structures can all be accommodated. We investigate the lattice of subvarieties of pointed left-residuated $$\ell $$ ℓ -groupoids, their ideals, and develop a theory of left nuclei. Finally, we extend some parts of the theory of join-completions of residuated $$\ell $$ ℓ -groupoids to the left-residuated case, giving a new proof of MacLaren’s theorem for orthomodular lattices.

Splittings in varieties of logic

2021 ◽  
pp. 1-48
Author(s):  
Brian A. Davey ◽  
Tomasz Kowalski ◽  
Christopher J. Taylor
Keyword(s):  
Residuated Lattices ◽  
Heyting Algebras ◽  
General Lemma ◽  
Negative Results ◽  
Splitting Algebras

We study splittings or lack of them, in lattices of subvarieties of some logic-related varieties. We present a general lemma, the non-splitting lemma, which when combined with some variety-specific constructions, yields each of our negative results: the variety of commutative integral residuated lattices contains no splitting algebras, and in the varieties of double Heyting algebras, dually pseudocomplemented Heyting algebras and regular double [Formula: see text]-algebras the only splitting algebras are the two-element and three-element chains.

Forbidden structures in Heyting algebras with respect to sublattices

2017 ◽  
Vol 11 ◽  
pp. 211-224
Author(s):  
Manish Agalave ◽  
R. S. Shewale ◽  
Vilas Kharat
Keyword(s):  
Heyting Algebras

scholarly journals Bitopological duality for distributive lattices and Heyting algebras

2010 ◽  
Vol 20 (3) ◽  
pp. 359-393 ◽  
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.

Interpretations into Heyting algebras

1987 ◽  
Vol 24 (1-2) ◽  
pp. 149-166 ◽  
Author(s):  
Renato Lewin
Keyword(s):  
Heyting Algebras

Perfect residuated lattice ordered monoids

2010 ◽  
Vol 60 (6) ◽  
Author(s):  
Jiří Rachůnek ◽  
Dana Šalounová
Keyword(s):  
Probability Measure ◽  
Residuated Lattice ◽  
Residuated Lattices ◽  
Heyting Algebras ◽  
Effect Algebras ◽  
Quantum Logics ◽  
Mv Algebras ◽  
Intuitionistic Logics

AbstractBounded Rℓ-monoids form a large subclass of the class of residuated lattices which contains certain of algebras of fuzzy and intuitionistic logics, such as GMV-algebras (= pseudo-MV-algebras), pseudo-BL-algebras and Heyting algebras. Moreover, GMV-algebras and pseudo-BL-algebras can be recognized as special kinds of pseudo-MV-effect algebras and pseudo-weak MV-effect algebras, i.e., as algebras of some quantum logics. In the paper, bipartite, local and perfect Rℓ-monoids are investigated and it is shown that every good perfect Rℓ-monoid has a state (= an analogue of probability measure).

Subalgebras of closure algebras

1992 ◽  
Vol 25 (3) ◽  
pp. 205-219 ◽  
Author(s):  
G. Hansoul ◽  
L. Vrancken-Mawet
Keyword(s):  
Closure Algebras
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