Algebra universalis

, Volume 64, Issue 3–4, pp 313–338 | Cite as

Graphical algebras — a new approach to congruence lattices

  • Toby KenneyEmail author


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.

2010 Mathematics Subject Classification

Primary: 08A30 Secondary: 06B15 

Keywords and phrases

congruence lattice graphical composition graphical algebra 


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Copyright information

© Springer Basel AG 2011

Authors and Affiliations

  1. 1.Ústav vedy a výskumuUniverzita Mateja BelaBanská BystricaSlovakia

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