Quasi- and pseudo-inverses of monotone functions, and the construction of t-norms. (English) Zbl 0953.26008

Summary: Several properties of quasi- and pseudo-inverses of a non-decreasing real function are discussed. Based on a result of Schweizer and Sklar, for a given triangular norm \(T\) and non-decreasing function \(f\) a construction method leading to a commutative, fully ordered semigroup on the unit interval is given. A similar construction based on the pseudo-inverse implies that the resulting operation will be bounded from above by the minimum, but then the associativity may be violated. Several sufficient conditions for constructing new t-norms from a given one and a non-decreasing function \(f\), based on its quasi-inverse and on its pseudo-inverse, respectively, are discussed, together with illustrative examples.


26E50 Fuzzy real analysis
03E72 Theory of fuzzy sets, etc.
Full Text: DOI


[1] Demant, B., Deformation von \(t\)-Normen, ihre Symmetrien und Symmetriebrechungen, (Proc. Workshop Fuzzy Systeme ’94, Klassifikation, Entscheidungssupport & Control (1994), Bonn Deutsche Informatik-Akademie: Bonn Deutsche Informatik-Akademie München)
[2] Drossos, C.; Navara, M., Generalized \(t\)-conorms and closure operators, (Proc. EUFIT ’96. Proc. EUFIT ’96, Aachen (1996)), 22-26
[3] Faucett, W. M., Compact semigroups irreducibly connected between two idempotents, (Proc. Amer. Math. Soc., 6 (1955)), 741-747 · Zbl 0065.25204
[4] Jenei, S., Fibred triangular norms, Fuzzy Sets and Systems, 103, 67-82 (1999) · Zbl 0946.26017
[5] Klement, E. P.; Mesiar, R.; Pap, E., Additive generators of \(t\)-norms which are not necessarily continuous, (Proc. EUFIT ’96, Aachen, vol. 1 (1996)), 70-73
[6] E.P. Klement, R. Mesiar, E. Pap, Triangular Norms, in preparation.; E.P. Klement, R. Mesiar, E. Pap, Triangular Norms, in preparation. · Zbl 0972.03002
[7] Ling, C. M., Representation of associative functions, Publ. Math. Debrecen, 12, 189-212 (1965) · Zbl 0137.26401
[8] Mesiar, R., On some constructions of new triangular norms, Mathware Soft Comput., 2, 39-45 (1995) · Zbl 0837.47056
[9] 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
[10] Schweizer, B.; Sklar, A., Associative functions and abstract semigroups, Publ. Math. Debrecen, 10, 69-81 (1963) · Zbl 0119.14001
[11] Schweizer, B.; Sklar, A., Probabilistic Metric Spaces (1983), North-Holland: North-Holland Amsterdam · Zbl 0546.60010
[12] Viceník, P., A note to a construction of t-norms based on pseudo-inverses of monotone functions, Fuzzy Sets and Systems, 104, 15-18 (1999) · Zbl 0953.26009
[13] Viceník, P., A note to the generators of \(t\)-norms, BUSEFAL, 75, 33-38 (1998)
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