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Abstract:

A symmetric residuated lattice is an algebra A = (A, ∨, ∧, *, →, ∼, 1, 0) such that (A, ∨, ∧, *, →, 1, 0) is a commutative integral bounded residuated lattice and the equations ∼ ∼ x = x and ∼ (x ∨ y) = ∼ x ∧ ∼ y are satisfied. The aim of the paper is to investigate the properties of the unary operation ε defined by the prescription ε x = ∼ x → 0. We give necessary and sufficient conditions for ε being an interior operator. Since these conditions are rather restrictive (for instance, on a symmetric Heyting algebra ε is an interior operator if and only the equation (x → 0) ∨ ((x → 0) → 0) = 1 is satisfied) we consider when an iteration of ε is an interior operator. In particular we consider the chain of varieties of symmetric residuated lattices such that the n iteration of ε is a boolean interior operator. For instance, we show that these varieties are semisimple. When n = 1, we obtain the variety of symmetric stonean residuated lattices. We also characterize the subvarieties admitting representations as subdirect products of chains. These results generalize and in many cases also simplify, results existing in the literature. © 2009 Elsevier B.V. All rights reserved.

Registro:

Documento: Artículo
Título:Commutative integral bounded residuated lattices with an added involution
Autor:Cignoli, R.; Esteva, F.
Filiación:Departamento de Matemática, Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires, Ciudad Universitaria, 1428 Buenos Aires, Argentina
Institut d'Investigació en Intel.ligència Artificial (IIIA), Spanish National Research Council (CSIC), Campus UAB s/n, 08193 Bellaterra, Catalonia, Spain
Palabras clave:Interior operators; Order reversing involutions; Pseudocomplemented residuated lattices; Residuated lattices; Stonean residuated lattices
Año:2009
Volumen:161
Número:2
Página de inicio:150
Página de fin:160
DOI: http://dx.doi.org/10.1016/j.apal.2009.05.008
Título revista:Annals of Pure and Applied Logic
Título revista abreviado:Ann. Pure Appl. Logic
ISSN:01680072
CODEN:APALD
PDF:https://bibliotecadigital.exactas.uba.ar/download/paper/paper_01680072_v161_n2_p150_Cignoli.pdf
Registro:https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_01680072_v161_n2_p150_Cignoli

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Citas:

---------- APA ----------
Cignoli, R. & Esteva, F. (2009) . Commutative integral bounded residuated lattices with an added involution. Annals of Pure and Applied Logic, 161(2), 150-160.
http://dx.doi.org/10.1016/j.apal.2009.05.008
---------- CHICAGO ----------
Cignoli, R., Esteva, F. "Commutative integral bounded residuated lattices with an added involution" . Annals of Pure and Applied Logic 161, no. 2 (2009) : 150-160.
http://dx.doi.org/10.1016/j.apal.2009.05.008
---------- MLA ----------
Cignoli, R., Esteva, F. "Commutative integral bounded residuated lattices with an added involution" . Annals of Pure and Applied Logic, vol. 161, no. 2, 2009, pp. 150-160.
http://dx.doi.org/10.1016/j.apal.2009.05.008
---------- VANCOUVER ----------
Cignoli, R., Esteva, F. Commutative integral bounded residuated lattices with an added involution. Ann. Pure Appl. Logic. 2009;161(2):150-160.
http://dx.doi.org/10.1016/j.apal.2009.05.008