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Minimal varieties and quasivarieties

  • Clifford Bergman (a1) and Ralph McKenzie (a2)

Abstract

We prove that every locally finite, congruence modular, minimal variety is minimal as a quasivariety. We also construct all finite, strictly simple algebras generating a congruence distributive variety, such that the sett of unary term perations forms a group. Lastly, these results are applied to a problem in algebraic logic to give a sufficient condition for a deductive system to be structurally complete.

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References

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[1]Andréka, H., Németi, I. and Sain, I., ‘Abstract Model Theoretic Approach to Algebraic Logic,’ preprint, 1984.
[2]Blok, W. and Pigozzi, D., Algebraizable Logics, Memoirs Amer. Math. Soc. 396, 1989.
[3]Freese, R. and McKenzie, R., Commutator Theory for Congruence Modular Varieties, London Mathematical Society Lecture Notes no. 125, Cambridge University Press, Cambridge, 1987.
[4]Henkin, L., Monk, D. and Tarski, A., Cylindric Algebras, Part I, North-Holland, Amsterdam, 1971.
[5]Jonsson, B., ‘Algebras whose congruence lattices are distributive,’ Math. Scan. 21 (1967), 110121.
[6]Németi, I., ‘On varieties of cylindric algebras with applications to logic,’ Ann. Pure Appl. Logic 36 (1987), 235277.
[7]Szendrei, A., ‘Idempotent algebras with restrictions on subalgebras,’ Acta Sci. Math. 51 (1987), 251268.
[8]Szendrei, A., ‘Every indempotent plain algebra generates a minimal variety,’ Algebra Universalis 25 (1988), 3639.
[9]Szendrei, A., ‘Demi-primal algebras,’ Algebra Universalis 18 (1984), 117128.
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Journal of the Australian Mathematical Society
  • ISSN: 1446-7887
  • EISSN: 1446-8107
  • URL: /core/journals/journal-of-the-australian-mathematical-society
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