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    Wyświetlanie 1-9 z 9
    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:
    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:
    B-almost distributive fuzzy lattice
    Autorzy:
    Assaye, Berhanu
    Alemneh, Mihret
    Tefera, Gerima
    Powiązania:
    https://bibliotekanauki.pl/articles/749988.pdf
    Data publikacji:
    2018
    Wydawca:
    Uniwersytet Łódzki. Wydawnictwo Uniwersytetu Łódzkiego
    Tematy:
    almost distributive lattice
    Heyting almost distributive lattice (HADL)
    B-almost distributive lattice
    principal ideals of an almost distributive lattice
    Opis:
    The paper introduces the concept of B-Almost distributive fuzzy lattice (BADFL) in terms of its principal ideal fuzzy lattice. Necessary and sufficient conditions for an ADFL to become a B-ADFL are investigated. We also prove the equivalency of B-algebra and B-fuzzy algebra. In addition, we extend PSADL to PSADFL and prove that B-ADFL implies PSADFL.
    Źródło:
    Bulletin of the Section of Logic; 2018, 47, 3
    0138-0680
    2449-836X
    Pojawia się w:
    Bulletin of the Section of Logic
    Dostawca treści:
    Biblioteka Nauki
    Artykuł
      Wyświetlanie 1-9 z 9

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