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Lattices of Subsemivarieties of Certain Varieties

Published online by Cambridge University Press:  09 April 2009

Ahmad Shafaat
Ahmad Shafaat
Affiliation:
University of ManitobaWinnipeg, Manitoba, Canada
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For all positive integers m, n, m ≤ n, let m,n denote the variety of algebras with m n-ary operations ω1, …, ωm and n m-ary operations φ1, …, φn satisfying the system of identities The varieties m,n are considered by Swierczkowski [1] and by Akataeb and Smirnov [2]. Jonsson and Tarski [3] consider m,n in the case m = 1, n = 2. In [2] it is shown that the lattice Lm,n of subvarieties of m,n, is uncountable except when m = 1 in which case Lm,n has a very simple description. In particular, for n > 1 the lattice L1,n is the two element chain which means that 1,n has no proper subvarieties.

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 12 , Issue 1 , February 1971 , pp. 15 - 20
Copyright
Copyright © Australian Mathematical Society 1971

References

[1]Swierczkowski, S., ‘On isomorphic free algebras’, Fund. Math. 50 (1961), 35–44.CrossRefGoogle Scholar
[2]Akataeb, A. A. and Smirnov, D. M., ‘Lattices of subvarieties of varieties of algebras’ [Russian] Algebra i Logica 7 (1968), 5–25.Google Scholar
[3]Jonsson, Bjarni and Tarski, Alfred, ‘On two properties of free algebras’, Math. Scand. 9 (1961), 95–101.CrossRefGoogle Scholar
[4]Shafaat, Ahmad, ‘Characterizations of some universal classes of algebras’, J. London Math. Soc. 2 (1970), 385–388.CrossRefGoogle Scholar