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Subdirect products of certain varieties of unary algebras. (English) Zbl 1174.08301

Summary: J.Płonka in [Simon Stevin 59, 353–364 (1985; Zbl 0591.08006)] noted that one could expect that the regularization \({\mathcal R}(\mathbf K)\) of a variety \({\mathbf K}\) of unary algebras is a subdirect product of \({\mathbf K}\) and the variety \({\mathbf D}\) of all discrete algebras (unary semilattices), but this is not the case. The purpose of this note is to show that his expectation is fulfilled for those and only those irregular varieties \({\mathbf K}\) which are contained in the generalized variety \(\mathbf {TDir}\) of the so-called trap-directable algebras.

MSC:

08A60 Unary algebras
08B15 Lattices of varieties
08B26 Subdirect products and subdirect irreducibility

Citations:

Zbl 0591.08006
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References:

[1] C. J. Ash: Pseudovarieties, generalized varieties and similarly described classes. J. Algebra 92 (1985), 104–115. · Zbl 0548.08007 · doi:10.1016/0021-8693(85)90147-4
[2] S. Bogdanović, M. Ćirić, B. Imreh, T. Petković, and M. Steinby: On local properties of unary algebras. Algebra Colloquium 10 (2003), 461–478. · Zbl 1032.08002
[3] S. Bogdanović, M. Ćirić, and T. Petković: Generalized varieties of algebras. Internat. J. Algebra Comput. Submitted.
[4] S. Bogdanović, M. Ćirić, T. Petković, B. Imreh, and M. Steinby: Traps, cores, extensions and subdirect decompositions of unary algebras. Fundamenta Informaticae 34 (1999), 51–60. · Zbl 0935.68058
[5] S. Bogdanović, B. Imreh, M. Ćirić, and T. Petković: Directable automata and their generalizations. A survey. Novi Sad J. Math. 29 (1999), 31–74.
[6] S. Burris, H.P. Sankappanavar: A Course in Universal Algebra. Springer-Verlag, New York, 1981. · Zbl 0478.08001
[7] M. Ćirić, S. Bogdanović: Lattices of subautomata and direct sum decompositions of automata. Algebra Colloquium 6 (1999), 71–88. · Zbl 0943.68117
[8] F. Gécseg, I. Peák: Algebraic Theory of Automata. Akadémiai Kiadó, Budapest, 1971.
[9] G. Grätzer: Universal Algebra, 2nd ed. Springer-Verlag, New York-Heidelberg-Berlin, 1979.
[10] T. Petković, M. Ćirić, and S. Bogdanović: Decompositions of automata and transition semigroups. Acta Cybernetica (Szeged) 13 (1998), 385–403. · Zbl 0926.68079
[11] J. Płonka: On the sum of a system of disjoint unary algebras corresponding to a given type. Bull. Acad. Pol. Sci., Ser. Sci. Math. 30 (1982), 305–309. · Zbl 0506.08004
[12] J. Płonka: On the lattice of varieties of unary algebras. Simon Stevin 59 (1985), 353–364. · Zbl 0591.08006
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