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Reflections on and of minor-closed classes of multisorted operations. (English) Zbl 1406.08002

The Galois theory for minor-closed sets of functions developed by N. Pippenger [Discrete Math. 254, No. 1–3, 405–419 (2002; Zbl 1010.06012)] is extended to multisorted functions in the sense of W. Wechler [Universal algebra for computer scientists. Berlin etc.: Springer-Verlag (1992; Zbl 0748.68002)]. While most of the existing theory translates easily, some multisorted peculiarities require special attention, in particular with regard to reflections in the sense of L. Barto et al. [Isr. J. Math. 223, 363–398 (2018; Zbl 1397.08002)].

MSC:

08A68 Heterogeneous algebras
06A15 Galois correspondences, closure operators (in relation to ordered sets)
03C05 Equational classes, universal algebra in model theory
08A40 Operations and polynomials in algebraic structures, primal algebras
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References:

[1] Barto, L.; Opršal, J.; Pinsker, M., The wonderland of reflections, Israel J. Math., 223, 363-398, (2018) · Zbl 1397.08002
[2] Couceiro, M.; Foldes, S., On closed sets of relational constraints and classes of functions closed under variable substitutions, Algebra Universalis, 54, 149-165, (2005) · Zbl 1095.08002
[3] Lau, D.: Function Algebras on Finite Sets. A Basic Course on Many-Valued Logic and Clone Theory. Springer, Heidelberg (2006) · Zbl 1105.08001
[4] Lehtonen, E., Pöschel, R., Waldhauser, T.: Reflection-closed varieties of multisorted algebras and minor identities. Algebra Univers. https://doi.org/10.1007/s00012-018-0547-3
[5] Lehtonen, E., Waldhauser, T.: Minor posets of functions as quotients of partition lattices. Order (2018). https://doi.org/10.1007/s11083-018-9453-8
[6] Pippenger, N., Galois theory for minors of finite functions, Discrete Math., 254, 405-419, (2002) · Zbl 1010.06012
[7] Pöschel, R.: Galois connections for operations and relations. In: Denecke, K., Erné, M., Wismath, S.L. (eds.) Galois connections and applications, Math. Appl., vol. 565, pp. 231-258. Kluwer Academic Publishers, Dordrecht (2004) · Zbl 1063.08003
[8] Pöschel, R., Kalužnin, L.A.: Funktionen- und Relationenalgebren. Ein Kapitel der diskreten Mathematik. VEB Deutscher Verlag der Wissenschaften, Berlin (1979)
[9] Wechler, W.: Universal Algebra for Computer Scientists, EATCS Monogr. Theoret. Comput. Sci., vol. 25. Springer, Berlin (1992) · Zbl 0748.68002
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