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Lehtonen, Erkko; Pöschel, Reinhard; Waldhauser, Tamás

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

Algebra Univers. 79, No. 3, Paper No. 71, 19 p. (2018).
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)].
Reviewer: Manfred Armbrust (Köln)

Cited in 1 Document

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

Keywords:

minor of function; multisorted function; multisorted operation; reflection

Citations:

Zbl 1010.06012; Zbl 0748.68002; Zbl 1397.08002
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\textit{E. Lehtonen} et al., Algebra Univers. 79, No. 3, Paper No. 71, 19 p. (2018; Zbl 1406.08002)
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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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