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Generalized continuous and left-continuous t-norms arising from algebraic semantics for fuzzy logics

AuthorsNoguera, Carles CSIC ORCID; Esteva, Francesc CSIC ORCID ; Godo, Lluis CSIC ORCID
KeywordsLeft-continuous t-norms
Mathematical fuzzy logic
Continuous t-norms
Rational chains
Monoidal t-norm based logic
Issue Date2010
CitationInformation Sciences 180: 1354- 1372 (2010)
AbstractThis paper focuses on the issue of how generalizations of continuous and left-continuous t-norms over linearly ordered sets should be from a logical point of view. Taking into account recent results in the scope of algebraic semantics for fuzzy logics over chains with a monoidal residuated operation, we advocate linearly ordered BL-algebras and MTL-algebras as adequate generalizations of continuous and left-continuous t-norms respectively. In both cases, the underlying basic structure is that of linearly ordered residuated lattices. Although the residuation property is equivalent to left-continuity in t-norms, continuous t-norms have received much more attention due to their simpler structure. We review their complete description in terms of ordinal sums and discuss the problem of describing the structure of their generalization to BL-chains. In particular we show the good behavior of BL-algebras over a finite or complete chain, and discuss the partial knowledge of rational BL-chains. Then we move to the general non-continuous case corresponding to left-continuous t-norms and MTL-chains. The unsolved problem of describing the structure of left-continuous t-norms is presented together with a fistful of construction-decomposition techniques that apply to some distinguished families of t-norms and, finally, we discuss the situation in the general study of MTL-chains as a natural generalization of left-continuous t-norms. © 2009 Elsevier Inc. All rights reserved.
Identifiersdoi: 10.1016/j.ins.2009.12.011
issn: 0020-0255
Appears in Collections:(IIIA) Artículos
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