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    Rodgers, J. D. 2013. Generalised Existence Varieties of Regular Semigroups. Communications in Algebra, Vol. 41, Issue. 10, p. 3969.
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    PIN, JEAN-ÉRIC and SILVA, PEDRO V. 2011. ON PROFINITE UNIFORM STRUCTURES DEFINED BY VARIETIES OF FINITE MONOIDS. International Journal of Algebra and Computation, Vol. 21, Issue. 01n02, p. 295.
    RODGERS, J. D. 2008. On E-pseudovarieties of finite regular semigroups. Bulletin of the Australian Mathematical Society, Vol. 77, Issue. 02,
    Gorbunov, V. A. 1996. Structure of lattices of varieties and lattices of quasivarieties: similarity and difference. III. Algebra and Logic, Vol. 34, Issue. 6, p. 359.
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Lattices of pseudovarieties

  • P. Agliano (a1) and J. B. Nation (a1)
Abstract

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.

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Copyright
COPYRIGHT: © Australian Mathematical Society 1989
References
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[1]Ash, C. J., ‘Pseudovarieties, generalized varieties and similarly described classes’, J. Algebra 92 (1985), 104–115.
[2]Almeida, J., ‘Power pseudovarieties of semigroups. I, II’, Semigroup Forum 33 (1986), 357–373; 375–390.
[3]Almeida, J., ‘The algebra of implicit operations’, preprint.
[4]Baker, K. A. and Hales, A. W., ‘From a lattice to its ideal lattice’, Algebra Universalis 4 (1974), 250–258.
[5]Baldwin, J. T. and Berman, J., ‘Varieties and finite closure conditions’, Colloq. Math. 35 (1976), 15–20.
[6]Banaschewski, B., ‘The Birkhoff Theorem for varieties of finite algebras’, Algebra Universalis 17 (1983), 360–368.
[7]Eilenberg, S. and Schützenberger, M. P., ‘On pseudovarieties’, Adv. in Math. 19 (1976), 413–418.
[8]Hall, T. E. and Johnston, K. G., ‘The lattice of pseudovarieties of inverse semigroups’, Pacific J. Math., to appear.
[9]Lampe, W. A., ‘A property of the lattice of equational theories’, Algebra Universalis 23 (1986), 61–69.
[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).
[11]Reiterman, J., ‘The Birkhoff theorem for finite algebras’, Algebra Universalis 14 (1982), 1–10.
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Journal of the Australian Mathematical Society
  • ISSN: 1446-7887
  • EISSN: 1446-8107
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