Abstract
In 1970, H. Werner considered the question of which sublattices of partition lattices are congruence lattices for an algebra on the underlying set of the partition lattices. He showed that a complete sublattice of a partition lattice is a congruence lattice if and only if it is closed under a new operation called graphical composition. We study the properties of this new operation, viewed as an operation on an abstract lattice. We obtain some necessary properties, and we also obtain some sufficient conditions for an operation on an abstract lattice L to be this operation on a congruence lattice isomorphic to L. We use this result to give a new proof of Grätzer and Schmidt’s result that any algebraic lattice occurs as a congruence lattice.
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Grätzer G., Schmidt E.T.: Characterizations of congruence lattices of abstract algebras. Acta Sci. Math. (Szeged) 24, 34–59 (1963)
Haiman M.: Proof theory for linear lattices. Advances in Mathematics 58, 209–242 (1985)
Jónsson B.: On the representation of lattices. Math. Scand. 1, 196–206 (1953)
Pálfy P.P., Pudlák P.: Congruence lattices of finite algebras and intervals in subgroup lattices of finite groups. Alg. Univ. 11, 22–27 (1980)
Pudlák P.: A new proof of the congruence lattice representation theorem. Alg. Univ. 6, 269–275 (1976)
Quackenbush R.W., Wolk B.: Strong representations of congruence lattices. Alg. Univ. 1, 165–166 (1971)
Repnitskiǐ V., Tůma J.: Intervals in subgroup lattices of countable locally finite groups. Alg. Univ. 59, 49–71 (2008)
Růzička P., Tůma J., Wehrung F.: Distributive congruence lattices of congruence-permutable algebras. J. Alg. 311, 96–116 (2007)
Snow J.W.: A constructive approach to the finite congruence lattice representation problem. Alg. Univ. 43, 279–293 (2000)
Snow J.W.: Almost distributive sublattices and congruence heredity. Alg. Univ. 57, 3–14 (2007)
Snow J.W.: OPC lattices and congruence heredity. Alg. Univ. 58, 59–71 (2008)
Tůma J.: Intervals in subgroup lattices of infinite groups. J. Alg. 125, 367–399 (1989)
Werner, H.: Which partition lattices are congruence lattices? Col. Math. Soc. Ján Bolyai 14, Lattice Theory, Szeged,(Hungary) 433–453, (1974)
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Presented by K. Kearnes.
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Kenney, T. Graphical algebras — a new approach to congruence lattices. Algebra Univers. 64, 313–338 (2010). https://doi.org/10.1007/s00012-011-0105-8
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DOI: https://doi.org/10.1007/s00012-011-0105-8