Representations through a monoid on the set of fuzzy implications

Fuzzy implications are one of the most important fuzzy logic connectives. In this work, we conduct a systematic algebraic study on the set II of all fuzzy implications. To this end, we propose a binary operation, denoted by ⊛, which makes (I,⊛I,⊛) a non-idempotent monoid. While this operation does not give a group structure, we determine the largest subgroup SS of this monoid and using its representation define a group action of SS that partitions II into equivalence classes. Based on these equivalence classes, we obtain a hitherto unknown representations of the two main families of fuzzy implications, viz., the f- and g-implications

Commutative integral bounded residuated lattices with an added involution

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.The research communicated in this paper was partially supported by a bilateral Argentinean - Spanish project CONICET - CSICPeer Reviewe

Commutative integral bounded residuated lattices with an added involution

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