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A Subsemilattice of the Lattice of Varieties of Lattice Ordered Groups

Published online by Cambridge University Press:  20 November 2018

Norman R. Reilly
Norman R. Reilly*
Affiliation:
Simon Fraser University, Burnaby, British Columbia
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Each variety of lattice ordered groups determines a variety of groups, namely the variety of groups generated by the groups i n . In this paper a completely new and different correspondence between varieties of groups and varieties of lattice ordered groups is developed. It is known that the variety of representable lattice ordered groups is defined by the law z+ Λ u-1z-u = 1. Here we consider the varieties defined by laws of this form where u is restricted to lie in some fully invariant subgroup of the free group Fx on a countable set X. All the varieties considered contain the variety of representable l-groups and therefore the free group with appropriate ordering.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 33 , Issue 6 , 01 December 1981 , pp. 1309 - 1318
Copyright
Copyright © Canadian Mathematical Society 1981

References

1. Adyan, S., Periodic groups of odd exponent, Proc. 2nd Internat. Conf. in the theory of groups, Canberra (1973), 8–12.Google Scholar
2. Bigard, A., Keimel, K. and Wolfenstein, S., Groupes et anneaux réticulé, Lecture notes in Math. (Springer-Verlag, 1977).Google Scholar
3. Conrad, P., Lattice ordered groups, Lecture notes, Tulane University (1970).Google Scholar
4. Glass, A. M. W., Ordered permutation groups, Bowling Green State University (1976).Google Scholar
5. Glass, A. M. W., Charles Holland, W. and McCleary, S. H., The structure of I-group varieties, Algebra Universalis 10 (1980), 1–20.Google Scholar
6. Holland, W. C., The lattice ordered group of automorphisms of an ordered set, Mich. Math. J. 10 (1963), 399–408.Google Scholar
7. Kopitov, V. M. and Medvedev, N. Y., About varieties of lattice ordered groups, Algebra and Logic 16 (1977), 417–423. (Russian)Google Scholar
8. Martinez, J., Free products in varieties of lattice ordered groups, Czech. Math. J. 22 (1972), 535–553.Google Scholar
9. Martinez, J., Varieties of lattice ordered groups, Math. Z. 137 (1974), 265–285.Google Scholar
10. Olshansky, A. U., On the problem of finite basis of identities in groups, Proc. Acad. Sci., USSR, Math. Sci. 34 (1970), 316–384.Google Scholar
11. Scrimger, E. B., A large class of small varieties of lattice ordered groups, Proc. Amer. Math. Soc. 51 (1975), 301–306.Google Scholar
12. Smith, J. E. H., The lattice of l-group varieties, Ph.D. thesis, Bowling Green State University (1976).Google Scholar