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Commutative integral bounded residuated lattices with an added involution

AuthorsCignoli, Roberto; Esteva, Francesc
KeywordsResiduated lattices
Stonean residuated lattices
Pseudocomplemented residuated lattices
Order reversing involutions
Interior operators
Issue Date2009
CitationAnnals of Pure and Applied Logic 161: 150- 160 (2009)
AbstractA 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.
Identifiersdoi: 10.1016/j.apal.2009.05.008
issn: 0168-0072
Appears in Collections:(IIIA) Artículos
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