On a variety of hemi-implicative semilattices

Castiglioni, José Luis; Fernández, Vı́ctor; Mallea, Héctor Federico; Martín, Hernán Javier San

doi:10.1007/s00500-022-06807-4

Cited by 3 publications

(13 citation statements)

References 8 publications

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“…We also give an equational base for the class of

\lbrace \wedge,\rightarrow,1\rbrace

‐subreducts of the algebras which belong to each one of the five subvarieties of

\mathsf {WH}

that we have considered in this paper. As we have mentioned before, the case of the class of

\lbrace \wedge,\rightarrow,1\rbrace

‐subreducts of subresiduated lattices was also studied in [6]. There it was proved, following a different technique than the one employed in the present paper, that this class is equal to the variety whose members are subresiduated semilattices.…”

Section: Introductionmentioning

confidence: 77%

“…Item (4) follows from items (1) and (2). Item (5) Proof. Throughout this proof we will use Corollary 3.6.…”

Section: 𝖨𝖲 Is the Variety Of Implicative Semilatticesmentioning

confidence: 99%

“…A similar argument to that employed in the proof of [14, Theorem 1] (cf. [6] for details) shows that if (𝐴, 𝐷) is a 𝑆𝑅𝑆-pair then (𝐴, ∧, →, 1) is a subresiduated semilattice and that if an algebra (𝐴, ∧, →, 1) of type (2, 2, 0) is a subresiduated semilattice then (𝐴, 𝐷) is a 𝑆𝑅𝑆-pair by defining 𝐷 = {𝑎 ∈ 𝐴 ∶ 1 → 𝑎 = 𝑎}.…”

Section: Representation For Subresiduated Semilatticesmentioning

confidence: 99%

“…In the proof of the following lemma we will use Remark 3.3. Part of its proof is like the proof of [5, Lemma 10]. However for the sake of completeness we opt to give a self‐contained proof of it.…”

Section: The Variety Of Weak Implicative Semilatticesmentioning

confidence: 99%

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On the implicative‐infimum subreducts of weak Heyting algebras

Celani,

San Martín

2024

Mathematical Logic Qtrly

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The variety of weak Heyting algebras was introduced in 2005 by Celani and Jansana. This corresponds to the strict implication fragment of the normal modal logic which is also known as the subintuitionistic local consequence of the class of all Kripke models. Subresiduated lattices are a generalization of Heyting algebras and particular cases of weak Heyting algebras. They were introduced during the 1970s by Epstein and Horn as an algebraic counterpart of some logics with strong implication previously studied by Lewy and Hacking. In this paper we study the class of implicative‐infimum subreducts of weak Heyting algebras. In particular, we prove that this class is a variety by giving an equational base for it. We also present a topological duality for the algebraic category whose objects are the implicative‐infimum subreducts of subresiduated lattices.

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“…We also give an equational base for the class of

\lbrace \wedge,\rightarrow,1\rbrace

‐subreducts of the algebras which belong to each one of the five subvarieties of

\mathsf {WH}

that we have considered in this paper. As we have mentioned before, the case of the class of

\lbrace \wedge,\rightarrow,1\rbrace

Section: Introductionmentioning

confidence: 77%

“…Item (4) follows from items (1) and (2). Item (5) Proof. Throughout this proof we will use Corollary 3.6.…”

Section: 𝖨𝖲 Is the Variety Of Implicative Semilatticesmentioning

confidence: 99%

Section: Representation For Subresiduated Semilatticesmentioning

confidence: 99%

Section: The Variety Of Weak Implicative Semilatticesmentioning

confidence: 99%

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On the implicative‐infimum subreducts of weak Heyting algebras

Celani,

San Martín

2024

Mathematical Logic Qtrly

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“…Más precisamente, probamos que toda álgebra de ShIS es isomorfa a una subálgebra de un miembro de ShIS, cuyo semiretículo subyacente es el de los crecientes de un poset. Parte de los resultados que exponemos en este capítulo fueron publicados en [8]…”

Section: Capítulounclassified

Un estudio sobre estructuras hemi-implicativas

Mallea

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La tesis se centra en el estudio de las estructuras hemi-implicativas y se divide en 5 capı́tulos. En el primer capı́tulo se presentan las variedades de retı́culos y semi-retı́culos hemi-implicativos, y se mencionan algunos ejemplos conocidos en la literatura de este tipo de estructuras. Entre estos ejemplos se destacan: Los semi-retı́culos implicativos. Las RW H-álgebras. Los retı́culos subresiduados. Las álgebras de semi-Heyting. Las álgebras de Hilbert con ı́nfimo. En el segundo capı́tulo se estudian algunos subreductos de los retı́culos subresiduados, en particular la cuasivariedad de álgebras de sub-Hilbert y la variedad de los semi-retı́culos subresiduados, luego se estudia una generalización de los retı́culos subresiduados donde el retı́culo subyacente no es necesariamente distributivo. Por último, se presenta una variedad que generaliza tanto a la variedad de las álgebras de Hilbert como a la de las álgebras de Hilbert con ı́nfimo, cuyos miembros llamamos semi-retı́culos de sub-Hilbert. En el tercer capı́tulo se estudia una representación de los retı́culos de sub- Hilbert por medio de ternas, una representación de los retı́culos de Hilbert por pares y luego se estudian las congruencias de los retı́culos de sub-Hilbert. En el cuarto capı́tulo se introduce una subvariedad propia de la variedad de los semi-retı́culos hemi-implicativos que contiene propiamente a cada una de las subvariedades estudiadas en esta tesis. Además, se presenta un teorema de representación para la variedad hallada. Por último, en el quinto capı́tulo se introduce una lógica proposicional algebrizable cuya semántica algebraica coincide con la cuasivariedad de álgebras de sub-Hilbert. Luego se estudian algunas expansiones de esta lógica con sus respectivas semánticas algebraicas y para algunas de ellas se estudia la propiedad de los modelos finitos.

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On self‐distributive weak Heyting algebras

Nourany,

Ghorbani,

Saeid

2023

Mathematical Logic Qtrly

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We use the left self‐distributive axiom to introduce and study a special class of weak Heyting algebras, called self‐distributive weak Heyting algebras (SDWH‐algebras). We present some useful properties of SDWH‐algebras and obtain some equivalent conditions of them. A characteristic of SDWH‐algebras of orders 3 and 4 is given. Finally, we study the relation between the variety of SDWH‐algebras and some of the known subvarieties of weak Heyting algebras such as the variety of Heyting algebras, the variety of basic algebras, the variety of subresiduated lattices, the variety of reflexive WH‐algebras (RWH‐algebras), and the variety of transitive WH‐algebras (TWH‐algebras).

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On a variety of hemi-implicative semilattices

Cited by 3 publications

References 8 publications

On the implicative‐infimum subreducts of weak Heyting algebras

On the implicative‐infimum subreducts of weak Heyting algebras

Un estudio sobre estructuras hemi-implicativas

On self‐distributive weak Heyting algebras

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