Abstract
Let L be any finite simple lattice of at least three elements, whose co-atoms intersect to 0. One principal result of the paper is that L is not dual isomorphic to the lattice of subvarieties of any locally finite variety. A second principal result is that these statements are equivalent: (i) L is isomorphic to the congruence lattice of a finite algebra with one basic operation; (ii) L is isomorphic either to the subspace lattice of a finite vector space, or for some permutation σ of a finite domain, to the lattice of equivalence relations invariant under σ.
Research supported by National Science Foundation grant MCS-8103455.
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References
R. Freese, W.A. Lampe and W. Taylor, Congruence lattices of algebras of fixed similarity type, I. Pacific J. Math. 82 (1979), 59–68.
B.Jónsson, Topics in Universal Algebra. Springer Lecture Notes, No. 250 (1970).
P.Köhler, M 7 as an interval in a subgroup lattice. Algebra Universalis (to appear).
W.A.Lampe, Congruence lattices of algebras of fixed similarity type, II. Pacific J. Math. (to appear).
R.McKenzie, A new product of algebras and a type reduction theorem. Algebra Universalis (to appear).
R.McKenzie, Categorical quasi-varieties revisited. (Preprint 1981).
P.P. Pálfy, On certain congruence lattices of finite unary algebras. Commentationes Math. Univ. Carolinae 19, 1 (1978), 89–85.
P.P.Pálfy, Unary polynomials in algebras, I. (Preprint 1982).
P.P. Pálfy and P. Pudlák, Congruence lattices of finite algebras and intervals in subgroup lattices of finite groups. Algebra Universalis 11 (1980), 22–27.
J. Płonka, Diagonal algebras. Fund. Math. 58 (1966), 309–321.
P. Pudlák and J. Tuma, Every finite lattice can be embedded into a finite partition lattice. Algebra Universalis 10 (1980), 74–95.
N.Sauer, M.G.Stone and R.H.Weedmark, Every finite algebra with congruence lattice M 7 has principal congruences, (this volume).
W. Taylor, The fine spectrum of a variety. Algebra Universalis 5 (1975), 262–303.
W. Taylor, Some applications of the term condition. Algebra Universalis 14 (1982), 11–25.
R. Wille, Eine Charakterisierung endlicher, ordnungspolynomvollständiger Verbände. Arch. Math. (Basel) 28 (1977), 557–560.
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© 1983 Springer-Verlag
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McKenzie, R. (1983). Finite forbidden lattices. In: Freese, R.S., Garcia, O.C. (eds) Universal Algebra and Lattice Theory. Lecture Notes in Mathematics, vol 1004. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0063438
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DOI: https://doi.org/10.1007/BFb0063438
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