×

On the clone of aggregation functions on bounded lattices. (English) Zbl 1390.06006

Summary: The main aim of this paper is to study aggregation functions on lattices via clone theory approach. Observing that the aggregation functions on lattices just correspond to \(0, 1\)-monotone clones, as the main result we show that for any finite \(n\)-element lattice \(L\) there is a set of at most \(2 n + 2\) aggregation functions on \(L\) from which the respective clone is generated. Namely, the set of generating aggregation functions consists only of at most \(n\) unary functions, at most \(n\) binary functions, and lattice operations \(\wedge, \vee\), and all aggregation functions of \(L\) are composed of them by usual term composition. Moreover, our approach works also for infinite lattices (such as mostly considered bounded real intervals \([a, b]\)), where in contrast to finite case infinite suprema (or, equivalently, a kind of limit process) have to be considered.

MSC:

06B05 Structure theory of lattices
06A15 Galois correspondences, closure operators (in relation to ordered sets)
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Baker, K. A.; Pixley, A. F., Polynomial interpolation and the Chinese remainder theorem for algebraic systems, Math. Zeitschrift, 143, 165-174 (1975) · Zbl 0292.08004
[2] Bandelt, H.-J., Retracts of hypercubes, J. Graph Theory, 8, 501-510 (1984) · Zbl 0551.05060
[3] Burris, S.; Sankappanavar, H. P., A Course in Universal Algebra (1981), Springer-Verlag · Zbl 0478.08001
[4] Csákány, B., Minimal clones - a minicourse, Algebra Univ., 54, 73-89 (2005) · Zbl 1088.08002
[5] De Baets, B.; Mesiar, R., Triangular norms on product lattices, Fuzzy Sets Syst., 104, 61-76 (1999) · Zbl 0935.03060
[6] Demirci, M., Aggregation operators on partially ordered sets and their categorical foundations, Kybernetika, 42, 261-277 (2006) · Zbl 1249.03091
[7] Grabisch, M.; Marichal, J.-L.; Mesiar, R.; Pap, E., Aggregation Functions (2009), Cambridge University Press: Cambridge University Press Cambridge
[8] Grätzer, G., Lattice Theory: Foundation (2011), Birkhäuser: Birkhäuser Basel · Zbl 1233.06001
[9] (Grätzer, G.; Wehrung, F., Lattice Theory: Special Topics and Applications, vol. 1 (2014), Birkhäuser, Basel) · Zbl 1296.06001
[10] Kaarli, K.; Pixley, A. F., Polynomial Completeness in Algebraic Systems (2001), Chapman & Hall/CRC: Chapman & Hall/CRC Boca Raton, Florida · Zbl 0964.08001
[11] Karacal, F.; Mesiar, R., Uninorms on bounded lattices, Fuzzy Sets Syst., 261, 33-43 (2015) · Zbl 1366.03229
[12] Kerkhoff, S.; Pöschel, R.; Schneider, F. M., A short introduction to clones, Electron. Notes Theor. Comput. Sci., 303, 107-120 (2014) · Zbl 1341.08003
[13] Komorníková, M.; Mesiar, R., Aggregation functions on bounded partially ordered sets and their classification, Fuzzy Sets Syst., 175, 48-56 (2011) · Zbl 1253.06004
[14] Lau, D., Function Algebras on Finite Sets (2006), Springer-Verlag: Springer-Verlag Berlin
[15] McKenzie, R.; McNulty, G.; Taylor, W., Algebras, Lattices and Varieties, vol. I (1987), Wadsworth & Brooks/Cole: Wadsworth & Brooks/Cole Monterey, California
[16] Post, E. L., The Two-Valued Iterative Systems of Mathematical Logic no. 5. The Two-Valued Iterative Systems of Mathematical Logic no. 5, Annals of Mathematics Studies (1941), Princeton University Press: Princeton University Press Princeton, NJ · Zbl 0063.06326
[17] Rosenberg, I. G., Über die funktionale Vollständigkeit in den mehrwertigen Logiken. Struktur der Funktionen von mehreren Veränderlichen auf endlichen Mengen, Rozpravy Československé Akad. Věd. Řada Mat. Přírod. Věd, 80, 3-93 (1970)
[18] Saminger-Platz, S.; Klement, E. P.; Mesiar, R., On extensions of triangular norms on bounded lattices, Indagat. Math., 19, 1, 135-150 (2008) · Zbl 1171.03011
[19] Tardos, G., A maximal clone of monotone operations which is not finitely generated, Order, 3, 211-218 (1986) · Zbl 0614.08006
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.