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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.

MSC:
26E50 Fuzzy real analysis
03E72 Theory of fuzzy sets, etc.
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[1] Demant, B., Deformation von t-normen, ihre symmetrien und symmetriebrechungen, ()
[2] Drossos, C.; Navara, M., Generalized t-conorms and closure operators, (), 22-26
[3] Faucett, W.M., Compact semigroups irreducibly connected between two idempotents, (), 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, (), 70-73
[6] 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 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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