Hostname: page-component-7b9c58cd5d-dkgms Total loading time: 0 Render date: 2025-03-13T22:31:02.233Z Has data issue: false hasContentIssue false
  • English
  • Français

Varieties generated by finite BCK-algebras

Published online by Cambridge University Press:  17 April 2009

William H. Cornish
William H. Cornish
Affiliation:
School of Mathematical Sciences, Flinders University, Bedford Park, South Australia 5042, Australia.
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Iséki's BCK-algebras form a quasivariety of groupoids and a finite BCK-algebra must satisfy the identity (En): xyn = xyn+1, for a suitable positive integer n. The class of BCK-algebras which satisfy (En) is a variety which has strongly equationally definable principal congruences, congruence-3-distributivity, and congruence-3-permutability. Thus, a finite BCK-algebra generates a 3-based variety of BCK-algebras. The variety of bounded commutative BCK-algebras which satisfy (En) is generated by n finite algebras, each of which is semiprimal.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 22 , Issue 3 , December 1980 , pp. 411 - 430
Copyright
Copyright © Australian Mathematical Society 1980

References

[1]Burris, Stanley, “On Baker's finite basis theorem for congruence distributive varieties”, Proc. Amer. Math. Soc. 73 (1979), 141–148.Google Scholar
[2]Cornish, William H., “A multiplier approach to implicative BCK-algebras”, Math. Sem. Notes Kobe Univ. 8 (1980), 157–169.Google Scholar
[3]Cornish, William H., “3-permutability and quasicommutative BCK-algebras”, Math. Japan, (to appear).Google Scholar
[4]Cornish, William H., “On positive implicative BCK-algebras”, Math. Sem. Notes Kobe Univ. (to appear).Google Scholar
[5]Day, Alan, “A note on the congruence extension property”, Algebra Universalis 1 (1971), 234–235.Google Scholar
[6]Foster, Alfred L. and Pixley, Alden F., “Semi-categorical algebras. II”, Math. Z. 85 (1964), 169–184.Google Scholar
[7]Fraser, Grant A. and Horn, Alfred, “Congruence relations in direct products”, Proc. Amer. Math. Soc. 26 (1970), 390–394.Google Scholar
[8]Fried, E., “A note on congruence extension property”, Acta Sci. Math. 40 (1978), 261–263.Google Scholar
[9]Hagemann, J. and Mitschke, A., “On n-permutable congruences”, Algebra Universalis 3 (1973), 8–12.Google Scholar
[10]Henkin, Leon, “An algebraic characterization of quantifiers”, Fund. Math. 37 (1950), 63–74.Google Scholar
[11]Iséki, Kiyoshi, “An algebra related with a propositional calculus”, Proc. Japan Acad. 42 (1966), 26–29.Google Scholar
[12]Iséki, Kiyoshi, “Topics of BCK-algebras”, Proceedings of the 1st Symposium on Semigroups, 44–56 (Shimane University, Shimane-Ken, Japan, 1977).Google Scholar
[13]Iséki, Kiyoshi, “On finite BCK-algebras”, Proceedings of the 3rd Symposium on Semigroups, 11–12 (Inter-University Seminar House, Kansai, 1979).Google Scholar
[14]Iséki, Kiyoshi and Tanaka, Shotaro, “Ideal theory of BCK-algebras”, Math. Japon. 21 (1976), 351–366.Google Scholar
[15]Iséki, Kiyoshi and Tanaka, Shotaro, “An introduction to the theory of BCK-algebras”, Math. Japon. 23 (1978), 1–26.Google Scholar
[16]Jónnson, Bjarni, “Algebras whose congruence lattices are distributive”, Math. Scand. 21 (1967), 110–121.Google Scholar
[17]Köhler, Peter and Pigozzi, Don, “Varieties with equationally definable principal congruences”, preprint.Google Scholar
[18]Komori, Yuichi, “The separation theorem of the ℵo-valued Lukasiewicz propositional logic”, Rep. Fac. Sci. Shizuoka Univ. 12 (1978), 1–5.Google Scholar
[19]Komori, Yuichi, “Super-Łukasiewicz implicational logics”, Nagoya Math. J. 72 (1978), 127–133.CrossRefGoogle Scholar
[20]Mitschke, Aleit, “Implication algebras are 3-permutable and 3-distributive”, Algebra Universalis 1 (1971), 182–186.Google Scholar
[21]Padmanabhan, R. and Quackenbush, R.W., “Equational theories of algebras with distributive congruences”, Proc. Amer. Math. Soc. 41 (1973), 373–377.Google Scholar
[22]Quackenbush, Robert W., “Primality: the influence of Boolean algebras in Universal Algebra”, Universal algebra, 2nd edition, Appendix 5, 401–416 (Springer-Verlag, New York, Heidelberg, Berlin, 1979).Google Scholar
[23]Romanowska, Anna and Traczyk, Tadeusz, “On commutative BCK-algebras”, preprint.Google Scholar
[24]Setō, Yasuo, “Some examples of BCK-algebras”, Math. Sem. Notes Kobe Univ. 5 (1977), 397–400.Google Scholar
[25]Shmuely, Zahava, “The structure of Galois connections”, Pacific J. Math. 54 (1974), no. 2, 209–225.Google Scholar
[26]Traczyk, Tadeusz, “On the variety of bounded commutative BCK-algebras”, Math. Japan. 24 (1979), 283–292.Google Scholar
[27]Yutani, Hiroshi, “Quasi-commutative BCK-algebras and congruence lattices”, Math. Sem. Notes Kobe Univ. 5 (1977), 469–480.Google Scholar