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How to construct left-continuous triangular norms – state of the art. (English) Zbl 1040.03021
Summary: Left-continuity of triangular norms is the characteristic property to make it a residuated lattice. Nowadays residuated lattices are subjects of intense investigation in the fields of universal algebra and nonclassical logic. The recently known construction methods resulting in left-continuous triangular norms are surveyed in this paper.

##### MSC:
 03B52 Fuzzy logic; logic of vagueness 03G10 Logical aspects of lattices and related structures
##### Keywords:
Residuated lattice; t-norm; Left-continuity; Construction
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##### References:
 [1] Budinčevič, M.; Kurilič, M., A family of strict and discontinuous triangular norms, Fuzzy sets and systems, 95, 381-384, (1998) · Zbl 0922.04006 [2] Cignoli, R.; Esteva, F.; Godo, L.; Montagna, F., On a class of left-continuous t-norms, Fuzzy sets and systems, 131, 283-296, (2002) · Zbl 1012.03032 [3] F. Esteva, L. Godo, QBL: towards a logic for left-continuous t-norms, Proc. of Joint EUSFLAT-ESTYLF’99 Conf. Palma de Mallorca, September 22-25, 1999, pp. 35-37. [4] Fodor, J.C., Contrapositive symmetry of fuzzy implications, Fuzzy sets and systems, 69, 141-156, (1995) · Zbl 0845.03007 [5] Frank, M.J., On the simultaneous associativity of F(x,y) and x+y−F(x,y), Aequationes math., 19, 194-226, (1979) · Zbl 0444.39003 [6] Gottwald, S., A treatise on many valued logics, studies in logic and computation, (2001), Research Studies Press Baldock, Hertfordshire, UK [7] Gottwald, S.; Jenei, S., A new axiomatization for involutive monoidal t-norm-based logic, Fuzzy sets and systems, 124, 303-308, (2001) · Zbl 0998.03024 [8] Hájek, P., Metamathematics of fuzzy logic, (1998), Kluwer Academic Publishers Dordrecht · Zbl 0937.03030 [9] Hájek, P., Observations on the monoidal t-norm logic, Fuzzy sets and systems, 132, 107-112, (2002) · Zbl 1012.03035 [10] Hőhle, U., Commutative, residuated l-monoids, (), 53-106 · Zbl 0838.06012 [11] Jenei, S., A characterization theorem on the rotation construction for triangular norms, Fuzzy sets and systems, 136, 283-289, (2003) · Zbl 1020.03021 [12] Jenei, S., A note on the ordinal sum theorem and its consequence for the construction of triangular norms, Fuzzy sets and systems, 126, 199-205, (2002) · Zbl 0996.03508 [13] Jenei, S., Fibred triangular norms, Fuzzy sets and systems, 103, 67-82, (1999) · Zbl 0946.26017 [14] Jenei, S., Generalized ordinal sum theorem and its consequence for the construction of triangular norms, Busefal, 80, 52-56, (1999) [15] Jenei, S., New family of triangular norms via contrapositive symmetrization of residuated implications, Fuzzy sets and systems, 110, 157-174, (1999) · Zbl 0941.03059 [16] Jenei, S., On the structure of rotation-invariant semigroups, Arch. math. logic, 42, 489-514, (2003) · Zbl 1028.06009 [17] Jenei, S., Structure of left-continuous triangular norms with strong associated negations. (I) rotation construction, J. appl. non-classical logics, 10, 83-92, (2000) · Zbl 1033.03512 [18] Jenei, S., Structure of left-continuous t-norms with strong associated negations. (II) rotation-annihilation construction, J. appl. non-classical logics, 11, 351-366, (2001) · Zbl 1037.03508 [19] S. Jenei, Structure of Girard monoids on [0,1], in: E.P. Klement, S.E. Rodabaugh (Eds.), Topological and Algebraic Structures in Fuzzy Sets, Kluwer Academic Publishers, Dordrecht, to appear. · Zbl 0993.68123 [20] S. Jenei, B. De Baets, Rotation and rotation-annihilation construction⧹newline of associative and partially compensatory aggregation operators, IEEE Trans. Fuzzy Systems, to appear. [21] Jenei, S.; Montagna, F., A general method for constructing left-continuous t-norms, Fuzzy sets and systems, 136, 263-282, (2003) · Zbl 1020.03020 [22] Jenei, S.; Montagna, F., A proof of standard completeness of esteva and Godo’s logic MTL, Studia logica, 70, 184-192, (2002) · Zbl 0997.03027 [23] Klement, E.P.; Mesiar, R.; Pap, E., Quasi- and pseudo-inverses of monotone functions and the construction of t-norms, Fuzzy sets and systems, 104, 3-13, (1999) · Zbl 0953.26008 [24] Klement, E.P.; Mesiar, R.; Pap, E., Triangular norms, (2000), Kluwer Academic Publishers Dordrecht · Zbl 0972.03002 [25] Klement, E.P.; Mesiar, R.; Pap, E., Triangular norms as ordinal sums of semigroups in the sense of A. H. Clifford, Semigroup forum, 65, 71-82, (2002) · Zbl 1007.20054 [26] Ling, C.-H., Representation of associative functions, Publ. math. debrecen, 12, 189-212, (1965) · Zbl 0137.26401 [27] Mostert, P.S.; Shields, A.L., On the structure of semigroups on a compact manifold with boundary, Ann. math., 65, 117-143, (1957) · Zbl 0096.01203 [28] Smutná, D., On a peculiar t-norm, Busefal, 75, 60-67, (1998) [29] Trillas, E., Sobre funciones de negación en la teorı́a de conjuntas difusos, Stochastica, 3, 47-60, (1979)
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