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Tytuł:
Semi-Heyting Algebras and Identities of Associative Type
Autorzy:
Cornejo, Juan M.
Sankappanavar, Hanamantagouda P.
Powiązania:
https://bibliotekanauki.pl/articles/749914.pdf
Data publikacji:
2019
Wydawca:
Uniwersytet Łódzki. Wydawnictwo Uniwersytetu Łódzkiego
Tematy:
semi-Heyting algebra
Heyting algebra
identity of associative type
subvariety of associative type
Opis:
An algebra A = ⟨A, ∨, ∧, →, 0, 1⟩ is a semi-Heyting algebra if ⟨A, ∨, ∧, 0, 1⟩ is a bounded lattice, and it satisfies the identities: x ∧ (x → y) ≈ x ∧ y, x ∧ (y → z) ≈ x ∧ [(x ∧ y) → (x ∧ z)], and x → x ≈ 1. ℋ denotes the variety of semi-Heyting algebras. Semi-Heyting algebras were introduced by the second author as an abstraction from Heyting algebras.  They share several important properties with Heyting algebras.  An identity of associative type of length 3 is a groupoid identity, both sides of which contain the same three (distinct) variables that occur in any order and that are grouped in one of the two (obvious) ways. A subvariety of ℋ is of associative type of length 3 if it is defined by a single identity of associative type of length 3. In this paper we describe all the distinct subvarieties of the variety ℋ of asociative type of length 3.  Our main result shows that there are 3 such subvarities of ℋ.
Źródło:
Bulletin of the Section of Logic; 2019, 48, 2
0138-0680
2449-836X
Pojawia się w:
Bulletin of the Section of Logic
Dostawca treści:
Biblioteka Nauki
Artykuł
Tytuł:
On the validity of the definition of a complement-classifier
Autorzy:
Stopa, Mariusz
Powiązania:
https://bibliotekanauki.pl/articles/1047622.pdf
Data publikacji:
2020-12-29
Wydawca:
Copernicus Center Press
Tematy:
category theory
topos theory
categorical logic
Heyting algebras
co-Heyting algebras
intuitionistic logic
dual to intuitionistic logic
complement-classifier
Opis:
It is well-established that topos theory is inherently connected with intuitionistic logic. In recent times several works appeared concerning so-called complement-toposes (co-toposes), which are allegedly connected to the dual to intuitionistic logic. In this paper I present this new notion, some of the motivations for it, and some of its consequences. Then, I argue that, assuming equivalence of certain two definitions of a topos, the concept of a complement-classifier (and thus of a co-topos as well) is, at least in general and within the conceptual framework of category theory, not appropriately defined. For this purpose, I first analyze the standard notion of a subobject classifier, show its connection with the representability of the functor Sub via the Yoneda lemma, recall some other properties of the internal structure of a topos and, based on these, I critically comment on the notion of a complement-classifier (and thus of a co-topos as well).
Źródło:
Zagadnienia Filozoficzne w Nauce; 2020, 69; 111-128
0867-8286
2451-0602
Pojawia się w:
Zagadnienia Filozoficzne w Nauce
Dostawca treści:
Biblioteka Nauki
Artykuł
Tytuł:
A Logic for Dually Hemimorphic Semi-Heyting Algebras and its Axiomatic Extensions
Autorzy:
Cornejo, Juan Manuel
Sankappanavar, Hanamantagouda P.
Powiązania:
https://bibliotekanauki.pl/articles/43189647.pdf
Data publikacji:
2022
Wydawca:
Uniwersytet Łódzki. Wydawnictwo Uniwersytetu Łódzkiego
Tematy:
semi-intuitionistic logic
dually hemimorphic semi-Heyting logic
dually quasi-De Morgan semi-Heyting logic
De Morgan semi-Heyting logic
dually pseudocomplemented semi-Heyting logic
regular dually quasi-De Morgan Stone semi-Heyting algebras of level 1
implicative logic
equivalent algebraic semantics
algebraizable logic
De Morgan Gödel logic
dually pseudocomplemented Gödel logic
Moisil's logic
3-valued Łukasiewicz logic
Opis:
The variety \(\mathbb{DHMSH}\) of dually hemimorphic semi-Heyting algebras was introduced in 2011 by the second author as an expansion of semi-Heyting algebras by a dual hemimorphism. In this paper, we focus on the variety \(\mathbb{DHMSH}\) from a logical point of view. The paper presents an extensive investigation of the logic corresponding to the variety of dually hemimorphic semi-Heyting algebras and of its axiomatic extensions, along with an equally extensive universal algebraic study of their corresponding algebraic semantics. Firstly, we present a Hilbert-style axiomatization of a new logic called "Dually hemimorphic semi-Heyting logic" (\(\mathcal{DHMSH}\), for short), as an expansion of semi-intuitionistic logic \(\mathcal{SI}\) (also called \(\mathcal{SH}\)) introduced by the first author by adding a weak negation (to be interpreted as a dual hemimorphism). We then prove that it is implicative in the sense of Rasiowa and that it is complete with respect to the variety \(\mathbb{DHMSH}\). It is deduced that the logic \(\mathcal{DHMSH}\) is algebraizable in the sense of Blok and Pigozzi, with the variety \(\mathbb{DHMSH}\) as its equivalent algebraic semantics and that the lattice of axiomatic extensions of \(\mathcal{DHMSH}\) is dually isomorphic to the lattice of subvarieties of \(\mathbb{DHMSH}\). A new axiomatization for Moisil's logic is also obtained. Secondly, we characterize the axiomatic extensions of \(\mathcal{DHMSH}\) in which the "Deduction Theorem" holds. Thirdly, we present several new logics, extending the logic \(\mathcal{DHMSH}\), corresponding to several important subvarieties of the variety \(\mathbb{DHMSH}\). These include logics corresponding to the varieties generated by two-element, three-element and some four-element dually quasi-De Morgan semi-Heyting algebras, as well as a new axiomatization for the 3-valued Łukasiewicz logic. Surprisingly, many of these logics turn out to be connexive logics, only a few of which are presented in this paper. Fourthly, we present axiomatizations for two infinite sequences of logics namely, De Morgan Gödel logics and dually pseudocomplemented Gödel logics. Fifthly, axiomatizations are also provided for logics corresponding to many subvarieties of regular dually quasi-De Morgan Stone semi-Heyting algebras, of regular De Morgan semi-Heyting algebras of level 1, and of JI-distributive semi-Heyting algebras of level 1. We conclude the paper with some open problems. Most of the logics considered in this paper are discriminator logics in the sense that they correspond to discriminator varieties. Some of them, just like the classical logic, are even primal in the sense that their corresponding varieties are generated by primal algebras.
Źródło:
Bulletin of the Section of Logic; 2022, 51, 4; 555-645
0138-0680
2449-836X
Pojawia się w:
Bulletin of the Section of Logic
Dostawca treści:
Biblioteka Nauki
Artykuł
Tytuł:
Equimorphy in varieties of double Heyting algebras
Autorzy:
Koubek, V.
Sichler, J.
Powiązania:
https://bibliotekanauki.pl/articles/966055.pdf
Data publikacji:
1998
Wydawca:
Polska Akademia Nauk. Instytut Matematyczny PAN
Tematy:
categorical universality
variety
double Heyting algebra
endomorphism monoid
equimorphy
Opis:
We show that any finitely generated variety V of double Heyting algebras is finitely determined, meaning that for some finite cardinal n(V), any class $\Cal S$ ⊆ V consisting of algebras with pairwise isomorphic endomorphism monoids has fewer than n(V) pairwise non-isomorphic members. This result complements the earlier established fact of categorical universality of the variety of all double Heyting algebras, and contrasts with categorical results concerning finitely generated varieties of distributive double p-algebras.
Źródło:
Colloquium Mathematicum; 1998, 77, 1; 41-58
0010-1354
Pojawia się w:
Colloquium Mathematicum
Dostawca treści:
Biblioteka Nauki
Artykuł
Tytuł:
Filozofia i logika intuicjonizmu
Autorzy:
Fila, Marlena
Powiązania:
https://bibliotekanauki.pl/articles/429258.pdf
Data publikacji:
2015
Wydawca:
Uniwersytet Papieski Jana Pawła II w Krakowie
Tematy:
intuitionism
axioms
matrices truth-
Heyting system
Gödel theorem about the inadequacy of finite dimensional matrices for Heyting system
infinite sequence of matrices
Opis:
At the end of the 19th century in the fundamentals of mathematics appeared a crisis. It was caused by the paradoxes found in Cantor’s set theory. One of the ideas a resolving the crisis was intuitionism – one of the constructivist trends in the philosophy of mathematics. Its creator was Brouwer, the main representative was Heyting. In this paper described will be attempt to construct a suitable logic for philosophical intuitionism theses. In second paragraph Heyting system will be present – its axioms and matrices truth-. Later Gödel theorem about the inadequacy of finite dimensional matrices for this system will be explained. At the end this paper an infinite sequence of matrices adequate for Heyting axioms proposed by Jaśkowski will be described.
Źródło:
Semina Scientiarum; 2015, 14
1644-3365
Pojawia się w:
Semina Scientiarum
Dostawca treści:
Biblioteka Nauki
Artykuł
Tytuł:
Preserving Filtering Unification by Adding Compatible Operations to Some Heyting Algebras
Autorzy:
Dzik, Wojciech
Radeleczki, Sándor
Powiązania:
https://bibliotekanauki.pl/articles/749994.pdf
Data publikacji:
2016
Wydawca:
Uniwersytet Łódzki. Wydawnictwo Uniwersytetu Łódzkiego
Tematy:
filtering unification
compatible operation
intuitionistic logic
Heyting algebra
residuated lattice
Opis:
We show that adding compatible operations to Heyting algebras and to commutative residuated lattices, both satisfying the Stone law ¬x ⋁ ¬¬x = 1, preserves filtering (or directed) unification, that is, the property that for every two unifiers there is a unifier more general then both of them. Contrary to that, often adding new operations to algebras results in changing the unification type. To prove the results we apply the theorems of [9] on direct products of l-algebras and filtering unification. We consider examples of frontal Heyting algebras, in particular Heyting algebras with the successor, γ and G operations as well as expansions of some commutative integral residuated lattices with successor operations.
Źródło:
Bulletin of the Section of Logic; 2016, 45, 3/4
0138-0680
2449-836X
Pojawia się w:
Bulletin of the Section of Logic
Dostawca treści:
Biblioteka Nauki
Artykuł
Tytuł:
Grzegorczyk Algebras Revisited
Autorzy:
Stronkowski, Michał M.
Powiązania:
https://bibliotekanauki.pl/articles/750038.pdf
Data publikacji:
2018
Wydawca:
Uniwersytet Łódzki. Wydawnictwo Uniwersytetu Łódzkiego
Tematy:
Grzegorczyk algebras
free Boolean extensions of Heyting algebras
stable homomorphisms
Opis:
We provide simple algebraic proofs of two important facts, due to Zakharyaschev and Esakia, about Grzegorczyk algebras.
Źródło:
Bulletin of the Section of Logic; 2018, 47, 2
0138-0680
2449-836X
Pojawia się w:
Bulletin of the Section of Logic
Dostawca treści:
Biblioteka Nauki
Artykuł
Tytuł:
On k-cyclic SHn-algebra
Autorzy:
Fernandez, A.
Powiązania:
https://bibliotekanauki.pl/articles/200167.pdf
Data publikacji:
2011
Wydawca:
Polska Akademia Nauk. Czytelnia Czasopism PAN
Tematy:
de Morgan algebra
Łukasiewicz algebras
Heyting algebra
Lattices and duality
Opis:
In this work we consider a new class of algebra called k-cyclic SHn-algebra (A, T) where A is an SHn-algebra and T is a lattice endomorphism such that Tk(x) = x, for all x, k is a positive integer. The main goal of this paper is to show a Priestley duality theorem for k-cyclic SHn-algebra.
Źródło:
Bulletin of the Polish Academy of Sciences. Technical Sciences; 2011, 59, 3; 303-304
0239-7528
Pojawia się w:
Bulletin of the Polish Academy of Sciences. Technical Sciences
Dostawca treści:
Biblioteka Nauki
Artykuł
Tytuł:
Quantum geometry, logic and probability
Autorzy:
Majid, Shahn
Powiązania:
https://bibliotekanauki.pl/articles/1047619.pdf
Data publikacji:
2020-12-29
Wydawca:
Copernicus Center Press
Tematy:
logic
noncommutative geometry
digital geometry
quantum gravity
duality
power set
Heyting algebra
Opis:
Quantum geometry on a discrete set means a directed graph with a weight associated to each arrow defining the quantum metric. However, these ‘lattice spacing’ weights do not have to be independent of the direction of the arrow. We use this greater freedom to give a quantum geometric interpretation of discrete Markov processes with transition probabilities as arrow weights, namely taking the diffusion form ∂+f = (−Δθ + q − p)f for the graph Laplacian Δθ, potential functions q, p built from the probabilities, and finite difference ∂+ in the time direction. Motivated by this new point of view, we introduce a ‘discrete Schrödinger process’ as ∂+ψ = ı(−Δ + V )ψ for the Laplacian associated to a bimodule connection such that the discrete evolution is unitary. We solve this explicitly for the 2-state graph, finding a 1-parameter family of such connections and an induced ‘generalised Markov process’ for f = |ψ|2 in which there is an additional source current built from ψ. We also mention our recent work on the quantum geometry of logic in ‘digital’ form over the field F2 = {0, 1}, including de Morgan duality and its possible generalisations.
Źródło:
Zagadnienia Filozoficzne w Nauce; 2020, 69; 191-236
0867-8286
2451-0602
Pojawia się w:
Zagadnienia Filozoficzne w Nauce
Dostawca treści:
Biblioteka Nauki
Artykuł
Tytuł:
Kilka uwag o przedmiocie logiki intuicjonistycznej
A few remarks on the subject of intuitionist logic
Einige Anmerkungen zum Gegenstand des intuitionistischen Aussagenkalküls
Autorzy:
Czernecka, Bożena
Powiązania:
https://bibliotekanauki.pl/articles/2016143.pdf
Data publikacji:
2001
Wydawca:
Katolicki Uniwersytet Lubelski Jana Pawła II. Towarzystwo Naukowe KUL
Tematy:
logika
filozofia logiki
filozofia matematyki
intuicjonizm
Heyting
logic
philosophy of logic
philosophy of mathematics
intuitionism
Opis:
Artykuł jest próbą odpowiedzi na pytanie, co jest przedmiotem logiki intuicjonistycznej. Rozważania opierają się na publikacjach twórców intuicjonizmu, zwłaszcza Heytinga. W tle poczyniono również komentarze na temat logiki klasycznej. Prawa klasycznej logiki zdań ustanawiają pewne obiektywne i jednocześnie najbardziej ogólne związki między faktami i zdarzeniami. Myśliciele zakładają istnienie takich powiązania, które charakteryzują stanowisko ontologiczne. Z drugiej strony prawa logiki intuicjonistycznej ustanawiają związki między konstrukcjami matematycznymi lub właściwościami tych konstrukcji na podstawie uprzednio utworzonych i zbadanych mentalnych konstrukcji matematycznych.
The article is an attempt to answer the question what is the subject of intuitionist logic. The considerations are based on publications by the creators of intuitionism, especially Heyting. In the background, comments were also made on classical logic. The laws of classical propositional logic establish certain objective and also the most general relationships between facts and events. Those thinkers assume the existence of such connections that characterize the ontological position. On the other hand, the laws of intuitionist logic establish relationships between mathematical constructions or the properties of these constructions on the basis of previously created and researched mental mathematical constructions.
In dem Artikel wird versucht, die Frage zu beantworten, was den Gegenstand des intuitionistischen Aussagenkalküls bildet. Die Erwägungen basieren auf den Veröffentlichungen von den Gründern des Intuitionismus, vor allem auf Heyting. Im Hintergrund werden auch Anmerkungen gemacht betreffend des Gegenstandes der klassischen Logik. Die Gesetze der klassischen Aussagenlogik stellen einige objektive und zugleich allgemeinste Zusammenhänge zwischen Sachverhalten, Tatsachen, Begebenheiten fest. Die Existenz, solchen Zusammenhänge nehmen die Denker an, welche die ontologische Stellungnahme kennzeichnet. Die Gesetze des intuitionistischen Aussagenkalküls dagegen stellen Zusammenhänge zwischen mathematischen Konstruktionen oder Eigenschaften dieser Konstruktionen auf Grund von den vorher gebildeten und untersuchten mentalen mathematischen Konstruktionen fest.
Źródło:
Roczniki Filozoficzne; 2001, 49, 1; 151-165
0035-7685
Pojawia się w:
Roczniki Filozoficzne
Dostawca treści:
Biblioteka Nauki
Artykuł
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