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A general method for constructing left-continuous t-norms. (English) Zbl 1020.03020
Summary: A new method for constructing (left-continuous) t-norms is introduced and analyzed in this paper. We construct via embedding a left-continuous t-norm from any countable residuated totally and densely ordered commutative integral monoid. Moreover, we can construct a left-continuous t-norm from any countable, totally ordered, commutative integral monoid which is not necessarily densely ordered and residuated. A special case, the embedding of such monoids on lexicographic product spaces, is investigated in detail, and several examples are demonstrated. The results shed some light on Chang’s MV-algebras, on a recently proposed ‘extraordinary’ t-norm, and on the standard semantics of the recently introduced logic \(\Pi\)-MTL of Hájek.

MSC:
03B52 Fuzzy logic; logic of vagueness
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[1] Budinc̆evic̆, M.; Kurilic̆, M., A family of strict and discontinuous triangular norms, Fuzzy sets and systems, 95, 381-384, (1998) · Zbl 0922.04006
[2] Chang, C.C., Algebraic analysis of many valued logics, Trans. amer. math. soc., 88, 467-490, (1958) · Zbl 0084.00704
[3] Climescu, A.C., Sur l’équation fonctionelle de l’associativité, Bull. école polytechn. iassy, 1, 1-16, (1946)
[4] Hájek, P., Metamathematics of fuzzy logic, (1998), Kluwer Academic Publishers Dordrecht · Zbl 0937.03030
[5] P. Hájek, Observations on the monoidal t-norm logic, Fuzzy Sets and Systems, to appear.
[6] 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
[7] Jenei, S., Structure of left-continuous t-norms with strong induced negations. (I) rotation construction, J. appl. non-classical logics, 10, 83-92, (2000) · Zbl 1033.03512
[8] 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
[9] Jenei, S.; Montagna, F., A proof of standard completeness of esteva and Godo’s monoidal logic MTL, Studia logica, 70, 184-192, (2002) · Zbl 0997.03027
[10] Klement, E.P.; Mesiar, R.; Pap, E., Triangular norms, (2000), Kluwer Academic Publishers Dordrecht · Zbl 0972.03002
[11] 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
[12] Smutná, D., On a peculiar t-norm, Busefal, 75, 60-67, (1998)
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