Article
Full entry |
PDF
(0.3 MB)
Feedback
PDF
(0.3 MB)
Feedback
Keywords:
unary algebra; subdirect product; variety; directable algebra
unary algebra; subdirect product; variety; directable algebra
Summary:
J. Płonka in [12] noted that one could expect that the regularization ${\mathcal R}(K)$ of a variety ${K}$ of unary algebras is a subdirect product of ${K}$ and the variety ${D}$ of all discrete algebras (unary semilattices), but is not the case. The purpose of this note is to show that his expectation is fulfilled for those and only those irregular varieties ${K}$ which are contained in the generalized variety ${TDir}$ of the so-called trap-directable algebras.
References:
[1] C. J. Ash: Pseudovarieties, generalized varieties and similarly described classes. J. Algebra 92 (1985), 104–115. MR 0772473 | Zbl 0548.08007
[2] S. Bogdanović, M. Ćirić, B. Imreh, T. Petković, and M. Steinby: On local properties of unary algebras. Algebra Colloquium 10 (2003), 461–478. MR 2013740
[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. DOI 10.3233/FI-1999-381205 | MR 1718110
[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. MR 1818327
[6] S. Burris, H. P. Sankappanavar: A Course in Universal Algebra. Springer-Verlag, New York, 1981. MR 0648287
[7] M. Ćirić, S. Bogdanović: Lattices of subautomata and direct sum decompositions of automata. Algebra Colloquium 6 (1999), 71–88. MR 1680653
[8] F. Gécseg, I. Peák: Algebraic Theory of Automata. Akadémiai Kiadó, Budapest, 1971. MR 0332374
[9] G. Grätzer: Universal Algebra, 2nd ed. Springer-Verlag, New York-Heidelberg-Berlin, 1979. MR 0538623
[10] T. Petković, M. Ćirić, and S. Bogdanović: Decompositions of automata and transition semigroups. Acta Cybernetica (Szeged) 13 (1998), 385–403. MR 1681152
[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. MR 0707740
[12] J. Płonka: On the lattice of varieties of unary algebras. Simon Stevin 59 (1985), 353–364. MR 0840857
Search
Partner of
