×

zbMATH — the first resource for mathematics

Archimedean components of triangular norms. (English) Zbl 1087.20041
A triangular norm (shortly t-norm) \(T\) is a binary operation on the real closed interval \([0,1]\) compatible with the natural order “\(\leq\)” of \([0,1]\) such that \(([0,1],T)\) is an Abelian, totally ordered semigroup having 1 as neutral element. Continuous t-norms are just ordinal sums of continuous Archimedean t-norms (turning \([0,1]\) into a topological semigroup).
In this paper the Archimedean components of triangular norms are studied. The main results are the following: A characterization of a function \(T\colon [0,1]^2\to[0,1]\) which is a continuous Archimedean t-norm. The result states that continuous Archimedean t-norms are characterized by having continuous additive generators. A characterization of a function \(T\colon[0,1]^2\to[0,1]\) which is continuous t-norm in terms of continuous Archimedean t-norms and ordinal sums. Each continuous t-norm is uniquely determined by its non-trivial Archimedean components. If \(T\) is a t-norm, then \(([0,1],T)\) is an ordinal sum of semigroups if and only if \(T\) is an ordinal sum of t-subnorms. If \(T\) is a t-norm, then the ordinal sum of its Archimedean components is the strongest t-norm which has the same Archimedean components as \(T\). Extensions of Archimedean components to triangular norms and a characterization of the class of triangular norms which are uniquely determined by their Archimedean components are given. Some construction methods for Archimedean components are given at the end.

MSC:
20M10 General structure theory for semigroups
06F05 Ordered semigroups and monoids
03E72 Theory of fuzzy sets, etc.
20M05 Free semigroups, generators and relations, word problems
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Klement, Mathematics of Fuzzy Sets. Logic, Topology, and Measure Theory pp 633– (1999) · doi:10.1007/978-1-4615-5079-2_13
[2] Kolesárová, BUSEFAL 80 pp 57– (1999)
[3] Fuchs, Partially ordered algebraic systems (1963) · Zbl 0137.02001
[4] Fodor, Fuzzy preference modelling and multicriteria decision support (1994) · Zbl 0827.90002 · doi:10.1007/978-94-017-1648-2
[5] DOI: 10.2307/2032928 · Zbl 0065.25204 · doi:10.2307/2032928
[6] De Baets, Technologies for constructing intelligent systems. 2: Tools pp 137– (2002) · doi:10.1007/978-3-7908-1796-6_11
[7] DOI: 10.2307/1993238 · Zbl 0082.02401 · doi:10.2307/1993238
[8] DOI: 10.2307/2372706 · Zbl 0055.01503 · doi:10.2307/2372706
[9] DOI: 10.1016/S0019-9958(65)90241-X · Zbl 0139.24606 · doi:10.1016/S0019-9958(65)90241-X
[10] Carruth, The theory of topological semigroups 75 (1983)
[11] Viceník, Topological and Algebraic Structures in Fuzzy Sets: A Handbook of Recent Developments in the Mathematics of Fuzzy Sets pp 441– (2003) · doi:10.1007/978-94-017-0231-7_20
[12] Butnariu, Triangular norm-based measures and games with fuzzy coalitions (1993) · Zbl 0804.90145 · doi:10.1007/978-94-017-3602-2
[13] DOI: 10.2307/2154504 · Zbl 0812.58062 · doi:10.2307/2154504
[14] Aczél, Lectures on functional equations and their applications (1966) · Zbl 0139.09301
[15] Schweizer, Probabilistic metric spaces (1983) · Zbl 0546.60010
[16] Schweizer, Pacific J. Math. 10 pp 313– (1960) · Zbl 0091.29801 · doi:10.2140/pjm.1960.10.313
[17] Klement, Illinois J. Math. 45 pp 1393– (2001)
[18] Klement, Triangular Norms (2000) · doi:10.1007/978-94-015-9540-7
[19] Kimberling, Publ. Math. Debrecen 20 pp 21– (1973)
[20] Jenei, J. Appl. Non-Classical Logics 10 pp 83– (2000) · Zbl 1033.03512 · doi:10.1080/11663081.2000.10510989
[21] Hofmann, Semigroup theory and its applications pp 15– (1996) · doi:10.1017/CBO9780511661877.004
[22] Hion, Izv. Akad. Nauk SSSR 21 pp 209– (1957)
[23] Hájek, Metamathematics of fuzzy logic (1998) · Zbl 0937.03030 · doi:10.1007/978-94-011-5300-3
[24] Hadžić, Fixed point theory in probabilistic metric spaces (2001) · doi:10.1007/978-94-017-1560-7
[25] Gottwald, A treatise on many-valued logic (2001) · Zbl 1048.03002
[26] Gierz, A Compendium of continuous lattices (1980) · Zbl 0452.06001 · doi:10.1007/978-3-642-67678-9
[27] Pap, Null-additive set functions (1995) · Zbl 0856.28001
[28] Paalman-de Miranda, Topological semigroups (1964) · Zbl 0136.26904
[29] Nelsen, An introduction to copulas (1999) · Zbl 0909.62052 · doi:10.1007/978-1-4757-3076-0
[30] Mostert, Ann. of Math., II. Ser. 65 pp 117– (1957)
[31] DOI: 10.1016/j.fss.2003.10.033 · Zbl 1043.03018 · doi:10.1016/j.fss.2003.10.033
[32] DOI: 10.1073/pnas.28.12.535 · Zbl 0063.03886 · doi:10.1073/pnas.28.12.535
[33] Ling, Publ. Math. Debrecen 12 pp 189– (1965)
[34] DOI: 10.1007/s002330010127 · Zbl 1007.20054 · doi:10.1007/s002330010127
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.