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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
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##### 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
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