A comparative study of adjoint triples

doi:10.1016/j.fss.2012.05.004

Fuzzy Sets and Systems

Volume 211, 16 January 2013, Pages 1-14

https://doi.org/10.1016/j.fss.2012.05.004 Get rights and content

Abstract

Adjoint triples are helpful as basic operators used in several domains. For example, adjoint triples play an important role in two important frameworks: multi-adjoint logic programming and multi-adjoint concept lattices. This paper shows that adjoint triples are an interesting generalization of t-norms and their residuated implications, since they preserve their main properties as well as they help to increase the flexibility of the operators used for computation in the considered framework. Furthermore, these operators will be related to other important general triples, specifically implication triples, and the definition of these latter operators will be improved.

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Citation Excerpt :
This logic is known as one of the most elegant axiomatizations of fuzzy logic, which has been taken into account as base logic for other developments, such as the logic for left-continuous t-norms presented by Esteva and Godo in [27], its extension considering right-continuous t-conorms [29], or the logics of subresiduated lattices from Epstein and Horn [26], among others. However, this axiomatization cannot be straightforwardly applied when more general and flexible algebras, such as multi-adjoint algebras [14–16], are considered. Another important goal of this paper is to enrich MLL with linguistic hedges, which are expressions such as “very”, “roughly”, “more or less” or “extremely”, which modify the meaning of statements (given in natural language) either strengthening or softening their impact.
This paper presents the multi-adjoint lattice logic (MLL) and its completeness and soundness. Specifically, the proposed many-valued propositional logic framework is defined on a multi-adjoint algebra whose underlying algebraic structure is a bounded lattice, which embeds the well-known basic logic (BL) given by Hájek on residuated lattices. The consideration of truth-stressing hedges in the multi-adjoint algebra is also studied and, as a consequence, two new logics extensions of MLL arise: the multi-adjoint lattice logic very true intensified (MLL $_{v t}$ ) and the multi-adjoint lattice logic ∨-very true intensified (MLL $_{\lor - v t}$ ). Finally, the soundness and completeness of the aforementioned logics are also proven.
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^☆: Partially supported by the Spanish Science Ministry under grant TIN2009-14562-C05-03 and by Junta de Andalucía under grant P09-FQM-5233.

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A comparative study of adjoint triples☆

Abstract

Fuzzy Sets Syst.

Ann. Pure Appl. Logic

Fuzzy Sets Syst.

Fuzzy Sets Syst.

Inf. Sci.

Inf. Sci.

Fuzzy Sets Syst.

Fuzzy Sets Syst.

Fuzzy Sets Syst.

Fuzzy Sets Syst.

Fuzzy Sets Syst.

Fuzzy Sets Syst.