Previous |  Up |  Next


unary algebra; subdirect product; variety; directable algebra
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.
[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. 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
Partner of
EuDML logo