Commutative integral bounded residuated lattices with an added involution

• Autores: Cignoli, R.; Esteva, F.
• Año de publicación: 2009
• Idioma: inglés
• Tipo de recurso: artículo
• Fuente: Ann. Pure Appl. Logic 2009;161(2):150-160
• Materia: Interior operators; Order reversing involutions; Pseudocomplemented residuated lattices; Residuated lattices; Stonean residuated lattices
• Institución: Universidad Nacional de Buenos Aires. Facultad de Ciencias Exactas y Naturales

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.