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Lattices of pseudovarieties

Part of: Varieties Other classes of algebra

Published online by Cambridge University Press:  09 April 2009

P. Agliano  and
J. B. Nation
P. Agliano
Affiliation:
Department of Mathematics, University of Hawaii at Manoa, Honolulu Hawaii 96822
J. B. Nation
Affiliation:
Department of Mathematics, University of Hawaii at Manoa, Honolulu Hawaii 96822
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Abstract

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We consider the lattice of pseudovarieties contained in a given pseudovariety P. It is shown that if the lattice L of subpseudovarieties of P has finite height, then L is isomorphic to the lattice of subvarieties of a locally finite variety. Thus not every finite lattice is isomorphic to a lattice of subpseudovarieties. Moreover, the lattice of subpseudovarieties of P satisfies every positive universal sentence holding in all lattice of subvarieties of varieties V(A) ganarated by algebras A ε P.

MSC classification

Secondary: 08B15: Lattices of varieties 08C99: None of the above, but in this section
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 46 , Issue 2 , April 1989 , pp. 177 - 183
Copyright
Copyright © Australian Mathematical Society 1989

References

[1]Ash, C. J., ‘Pseudovarieties, generalized varieties and similarly described classes’, J. Algebra 92 (1985), 104–115.CrossRefGoogle Scholar
[2]Almeida, J., ‘Power pseudovarieties of semigroups. I, II’, Semigroup Forum 33 (1986), 357–373; 375–390.CrossRefGoogle Scholar
[3]Almeida, J., ‘The algebra of implicit operations’, preprint.Google Scholar
[4]Baker, K. A. and Hales, A. W., ‘From a lattice to its ideal lattice’, Algebra Universalis 4 (1974), 250–258.Google Scholar
[5]Baldwin, J. T. and Berman, J., ‘Varieties and finite closure conditions’, Colloq. Math. 35 (1976), 15–20.Google Scholar
[6]Banaschewski, B., ‘The Birkhoff Theorem for varieties of finite algebras’, Algebra Universalis 17 (1983), 360–368.CrossRefGoogle Scholar
[7]Eilenberg, S. and Schützenberger, M. P., ‘On pseudovarieties’, Adv. in Math. 19 (1976), 413–418.CrossRefGoogle Scholar
[8]Hall, T. E. and Johnston, K. G., ‘The lattice of pseudovarieties of inverse semigroups’, Pacific J. Math., to appear.Google Scholar
[9]Lampe, W. A., ‘A property of the lattice of equational theories’, Algebra Universalis 23 (1986), 61–69.Google Scholar
[10]McKenzie, R., ‘Finite forbidden lattices’, Proceedings of the Universal Algebra and Lattice Theory Conference, Puebla 1982 (Lecture Notes in Math., vol. 1004, Springer-Verlag, pp. 176–205).CrossRefGoogle Scholar
[11]Reiterman, J., ‘The Birkhoff theorem for finite algebras’, Algebra Universalis 14 (1982), 1–10.CrossRefGoogle Scholar