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Quasivarieties and Varieties of Lattice-Ordered Groups

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Ordered Groups and Infinite Permutation Groups

Part of the book series: Mathematics and Its Applications ((MAIA,volume 354))

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Abstract

Many properties and statements of the theory of lattice-ordered groups (ℓ-groups) can be formulated and proved in terms of first order logic. Special mention should be made of properties expressed by universal sentences such as identities and implications, which can be referred to as the theory of varieties and quasivarieties, respectively, of ℓ-groups. The theory of varieties of ℓ-groups has been developed for more than two decades and contributions to it have been included in books and survey articles (Anderson and Feil [1], Kopytov and Medvedev [44, 45], Reily [63]). It was not until the mid-80s that the systematic investigation of the theory of quasivarieties of ℓ-groups began and the results obtained in this area are not available for wide audience yet. It is clear that the theory of quasivarieties is more general than that of varieties. Nevertheless, there have been obtained a number of results on quasivarieties of ℓ-groups asserting that helpful and non-trivial properties of t- groups can be defined by means of implications, and the theory of quasivarieties of ℓ-groups itself is exciting and profound.

This work was done with financial support of the Russian Fund of Fundamental Research (project code 93-011-1524)

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Author information

Authors and Affiliations

  1. Siberian Academy of Science, Mathematical Institute, Novosibirsk, 630090, Russia

    V. M. Kopytov

  2. Altai State University, Barnaul, 656099, Russia

    N. Ya. Medvedev

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  1. V. M. Kopytov
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  2. N. Ya. Medvedev
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Editors and Affiliations

  1. Bowling Green State University, USA

    W. Charles Holland

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© 1996 Kluwer Academic Publishers

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Kopytov, V.M., Medvedev, N.Y. (1996). Quasivarieties and Varieties of Lattice-Ordered Groups. In: Holland, W.C. (eds) Ordered Groups and Infinite Permutation Groups. Mathematics and Its Applications, vol 354. Springer, Boston, MA. https://doi.org/10.1007/978-1-4613-3443-9_1

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  • DOI: https://doi.org/10.1007/978-1-4613-3443-9_1

  • Publisher Name: Springer, Boston, MA

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