## On generating sets of the clone of aggregation functions on finite lattices.(English)Zbl 1446.06010

Summary: In a recent paper [the first and the third author, Inf. Sci. 329, 381–389 (2016; Zbl 1390.06006)] we have shown that aggregation functions on a bounded lattice $$L$$ form a clone, i.e., the set of functions closed under projections and composition of functions. Moreover, for any finite lattice $$L$$ we gave a finite set of unary and binary aggregation functions on $$L$$ from which the aggregation clone is generated. In this paper, a general method for constructing generating sets of the aggregation clone on $$L$$ is presented. Our approach is based on extending of $$L$$-valued capacities leading to so-called full systems of aggregation functions. Several full systems on $$L$$ are presented (including singleton ones) and their arities are discussed.

### MSC:

 06B05 Structure theory of lattices 06A15 Galois correspondences, closure operators (in relation to ordered sets) 08A40 Operations and polynomials in algebraic structures, primal algebras

Zbl 1390.06006
Full Text:

### References:

 [1] Anderson, I., Combinatorics of Finite Sets (2011), Dover Publications: Dover Publications Mineola [2] Beliakov, G.; Pradera, A.; Calvo, T., Studies fuzziness on soft computing, Aggregation Functions: A Guide for Practitioners, 221 (2007), Springer [3] Botur, M.; Halaš, R.; Mesiar, R.; Pócs, J., On generating of idempotent aggregation functions on finite lattices, Inf. Sci. (NY), 430-431, 39-45 (2018) [4] Bronevich, A.; Mesiar, R., Invariant continuous aggregation functions, Int. J. Gen. Syst., 39, 2, 177-188 (2010) · Zbl 1193.93009 [5] Burris, S.; Sankappanavar, H. P., A Course in Universal Algebra (1981), Springer-Verlag · Zbl 0478.08001 [6] Ghédira, K.; Dubuisson, B., Constrained Satisfaction Problems (2013), Wiley [7] Grabisch, M., Set Functions, Games and Capacities in Decision Making (2016), Springer Verlag: Springer Verlag Berlin · Zbl 1339.91003 [8] Grabisch, M.; Marichal, J. L.; Mesiar, R.; Pap, E., Aggregation Functions (2009), Cambridge University Press: Cambridge University Press Cambridge [9] Grätzer, G., Lattice Theory: Foundation (2011), Birkhäuser: Birkhäuser Basel · Zbl 1233.06001 [10] Halaš, R.; Mesiar, R.; Pócs, J., Generators of aggregation functions and fuzzy connectives, IEEE Trans. Fuzzy Syst., 24, 6, 1690-1694 (2016) [11] Halaš, R.; Pócs, J., On lattices with a smallest set of aggregation functions, Inf. Sci. (NY), 325, 316-323 (2015) · Zbl 1387.06006 [12] Halaš, R.; Pócs, J., On the clone of aggregation functions on bounded lattices, Inf. Sci. (NY), 329, 381-389 (2016) · Zbl 1390.06006 [13] Jablonskii, S. W.; Gawrilov, G. P.; Kudrjavcev, W. B., Boolesche funktionen und postsche klassen (1970), Braunschweig: Braunschweig Vieweg · Zbl 0193.29601 [14] Klement, E. P.; Mesiar, R., On the expected value of fuzzy events, Int. J. Uncertain. Fuzziness Knowl. Based Syst., 23, 57-74 (2015) · Zbl 1377.60012 [15] Lau, D., Function Algebras on Finite Sets (2006), Springer-Verlag: Springer-Verlag Berlin [16] Ovchinnikov, S.; Dukchovny, A., Integral representation of invariant functionals, J. Math. Anal. Appl., 244, 1, 228-232 (2000) · Zbl 0954.28010 [17] Torra, V.; Narukawa, Y., Modeling Decisions: Information Fusion and Aggregation Operators (2007), Springer-Verlag
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.