Abstract
Whereas there is a maximal proper variety ℜ of lattice-ordered groups, it is known that there is no maximal proper quasi-variety of lattice-ordered groups. We prove that there are 2ℵº (the maximal possible) pairwise incomparable quasi-varieties of lattice-ordered groups containing ℜ. Some of the distributive laws of the semigroup lattice of quasi-varieties are examined and their truth (or falsity) is established. It is also shown here that the latticeL of alll-group varieties is a sublattice of the latticeQ of quasi-varieties ofl-groups but fails to be a complete sublattice.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
S. I. Adyan,Infinite irreducible systems of group identities, Izv. Akad. Nauk SSR, Ser. Math. TOM34, No. 4 (1979), 715–734.
V. P. Belkin andV. A. Gorbunov,Filters in the lattices of quasi-varieties, Algebra and Logic,14, No. 4 (1975), 229–239.
G. Birkhoff,On the structure of abstract algebra, Proc. Cambridge Phil. Soc.,31 (1935), 433–454.
C. G. Chehata,An algebraically simple ordered group, Proc. London Math. Soc.,2 (1952), 183–197.
A. M. W.Glass,Ordered Permutation Groups, London Math. Soc., Lecture Notes Series 55, Cambridge University, Press.
- andYuri Gurevich,The word problem for lattice-orderded groups, Trans. Amer. Math. Soc., to appear.
A. M. W. Glass, W. C. Holland andS. H. McCleary,The structure of l-group varieties, Algebra Universal,10 (1980), 1–20.
W. C. Holland,The largest proper variety of lattice-ordered groups, Proc. Amer. Math. Soc.,57 (1976), 25–28.
Azriel Lévy,Basic Set Theory, Springer-Verlag, New York, 1979.
J. Martinez,Varieties of lattice-ordered groups, Math. Zeit.,137 (1974), 265–284.
N. Ya. Medvedev,The lattices of varieties of lattice-ordered groups and Lie algebras, Algebra and Logic,16, No. 4 (1977), 27–30 (English translation).
B. H. Neumann,On ordered groups, Amer. J. of Math.,71 (1949), 1–18.
N. R.Reilly,A subsemilattice of the lattice of varieties of lattice ordered groups, Canad. J. of Math., to appear.
— andR. Wroblewski,Suprema of classes of generalized Scrimger varieties of lattice-ordered groups, Math. Zeit.,176 (1981), 293–309.
J. E. Smith,The lattice of l-group varieties, Trans. Amer. Math. Soc.,257, No. 2 (1980), 347–357.
E. C. Weinberg,Free lattice-ordered abelian groups, Math. Ann.,151 (1963), 187–199.
Author information
Authors and Affiliations
Additional information
This article is a part of the author's Ph.D. dissertation which was directed by Professor A. M. W. Glass. The author wishes to express his sincere gratitude to Professor Glass for his assistance and encouragement during the writing of the dissertation and this article. He also wishes to thank Professor K. K. Hickin for his help with nilpotent material, in particular for his help in establishing Theorem 4.7.
Rights and permissions
About this article
Cite this article
Arora, A.K. Quasi-varieties of lattice ordered groups. Algebra Universalis 20, 34–50 (1985). https://doi.org/10.1007/BF01236804
Received:
Accepted:
Issue Date:
DOI: https://doi.org/10.1007/BF01236804