A property of the lattice of equational theories. (English) Zbl 0604.08003

Author’s summary: It has been conjectured that any algebraic lattice having a compact one could be represented as the lattice of equational theories extending some theory. However, we show that each lattice having such a representation satisfies a nontrivial quasidistributivity condition. In particular, \(M_ 3\) has no such representation.
Reviewer: E.Nelson


08B15 Lattices of varieties
06B15 Representation theory of lattices
03C05 Equational classes, universal algebra in model theory
Full Text: DOI


[1] S. Burris,On th structure of the lattice of equational classes of L(?), Algebra Universalis1 (1971) 39-45. · Zbl 0219.08001 · doi:10.1007/BF02944953
[2] S. Burris andE. Nelson,Embedding the dual of ? m in the lattice of equational classes of commutative semigroups, Proc. Amer. Math. Soc.30 (1971) 37-39.
[3] S. Burris andE. Nelson,Embedding the dual of ? m in the lattice of equational classes of semigroups, Algebra Universalis1 (1971) 248-254. · Zbl 0227.08006 · doi:10.1007/BF02944986
[4] A. Day andR. Freese,Characterizations of congruence modularity, I, Canadian J. Math.32 (1980) 1140-1167. · doi:10.4153/CJM-1980-087-6
[5] R. Freese, W. A. Lampe, andW. Taylor,Congruence lattices of algebras of fixed similarity type I, Pac. J. Math.82 (1979) 59-68. · Zbl 0394.06002
[6] V. A. Gorbunov,Quasi identities of two-element algebras, Algebra and Logic22 (1984) 83-88. (Algebra i Logika,22 (1981) 121-127.) · Zbl 0535.08006 · doi:10.1007/BF01978661
[7] J. Jezek,Intervals in the lattice of varieties, Alg. Univ.6 (1976) 147-158. · Zbl 0354.08007 · doi:10.1007/BF02485826
[8] J. Jezek,The lattice of equational theories, Part I, Part II, Part III; Czech. Math. J.31 (1981) 127-152;31 (1981) 573-603;32 (1982) 129-164. · Zbl 0477.08006
[9] J.Jezek,The lattice of equational theories, Part IV: Equational Theories of Finite Algebras (preprint). · Zbl 0477.08006
[10] B. Jonsson,Algebras whose congruence lattices are distributive, Math. Scand.21 (1967) 110-121. · Zbl 0167.28401
[11] S.Kogalovskii,Lattices of varieties, to appear. · Zbl 0906.08002
[12] W. A. Lampe,Congruence lattices of algebras of fixed similarity type, II, Pac. J. Math.103 (1982) 475-508. · Zbl 0511.08001
[13] W. A. Lampe,Lattices of subvarieties, I, Abstracts6 (1985), 85T-08-135, p. 254.
[14] W. A.Lampe,Lattices of subvarieties, I, preprint.
[15] A. I. Mal’cev,Problems on the border line of algebra and logic (Russian), Proc. Int. Cong, of Math. (Moscow 1966). Moscow 1968, MIR, 217-231. [English translation in A. I. Mal’cev,The metamathematics of algebraic systems. Collected papers: 1936-1967, North-Holland Pub. Co., Amsterdam (1971), 460-473.
[16] R. McKenzie,Definability in lattices of equational theories, Annals of Math. Logic3 (1971), 197-237. · Zbl 0328.02038 · doi:10.1016/0003-4843(71)90007-6
[17] R. McKenzie,Finite forbidden lattices, Proc. Fourth Int. Conf. on Universal Alg. and Lattice Theory, Puebla, 1982, Lecture Notes in Math. 1004 (1983), Springer-Verlag, Berlin, 176-205.
[18] G. McNulty,Structural diversity in the lattice of equational theories, Alg. Univ.13 (1981), 271-292. · Zbl 0485.08007 · doi:10.1007/BF02483841
[19] D.Pigozzi,The representation of certain abstract lattices as lattices of subvarieties, to appear.
[20] M. P. Tropin,Embedding a free lattice in a lattice of quasivarieties of distributive lattices with pseudocomplementation, Algebra and Logic22 (1984), 113-119. (Algebra i Logika22 (1983), 159-167. · Zbl 0541.06008 · doi:10.1007/BF01978664
[21] V. I. Tumanov,Finite distributive lattices of quasivarieties, Algebra and Logic22 (1984), 119-129. (Algebra i Logika22 (1983), 168-181). · Zbl 0548.08006 · doi:10.1007/BF01978665
[22] P. Whitman,Lattices, equivalence relations, and subgroups, Bull. Amer. Math. Soc.52 (1946) 507-522. · Zbl 0060.06505 · doi:10.1090/S0002-9904-1946-08602-4
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.