The use of fuzzy outranking relations in preference modelling
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Triangular norms on product lattices
Fuzzy Sets and Systems - Special issue on triangular norms
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On a class of left-continuous t-norms
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A general method for constructing left-continuous t-norms
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On Weakly Cancellative Fuzzy Logics
Journal of Logic and Computation
On the structure of left-continuous t-norms that have a continuous contour line
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The triple rotation method for constructing t-norms
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Strong non-standard completeness for fuzzy logics
Soft Computing - A Fusion of Foundations, Methodologies and Applications - Special issue (pp 315-357) "Ordered structures in many-valued logic"
Generalizations to the constructions of t-norms: Rotation(-annihilation) construction
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Cancellativity properties for t-norms and t-subnorms
Information Sciences: an International Journal
Residuated Lattices: An Algebraic Glimpse at Substructural Logics, Volume 151
Residuated Lattices: An Algebraic Glimpse at Substructural Logics, Volume 151
Rotation-invariant t-norms: Where triple rotation and rotation--annihilation meet
Fuzzy Sets and Systems
Triangular norms on partially ordered sets
Fuzzy Sets and Systems
Arithmetical Complexity of First-order Predicate Fuzzy Logics Over Distinguished Semantics
Journal of Logic and Computation
Pseudo-BCK algebras as partial algebras
Information Sciences: an International Journal
Systemic approach to fuzzy logic formalization for approximate reasoning
Information Sciences: an International Journal
States on finite monoidal t-norm based algebras
Information Sciences: an International Journal
Information Sciences: an International Journal
Triangular norm based graded convex fuzzy sets
Fuzzy Sets and Systems
Fuzzy adaptive synchronization of time-reversed chaotic systems via a new adaptive control strategy
Information Sciences: an International Journal
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This 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.