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Expansions of finite algebras and their congruence lattices

Abstract

In this paper, we present a novel approach to the construction of new finite algebras and describe the congruence lattices of these algebras. Given a finite algebra \({\langle B_0, \ldots \rangle}\), let \({B_1,B_2, \ldots , B_K}\) be sets that either intersect B 0 or intersect each other at certain points. We construct an overalgebra \({\langle A, FA \rangle}\), by which we mean an expansion of \({\langle B_0, \ldots \rangle}\) with universe \({A = B_0 \cup B_1 \cup \ldots \cup B_K}\), and a certain set F A of unary operations that includes mappings e i satisfying \({e^2_i = e_i}\) and e i (A) =  B i , for \({0 \leq i \leq K}\). We explore two such constructions and prove results about the shape of the new congruence lattices Con\({\langle A, F_A \rangle}\) that result. Thus, descriptions of some new classes of finitely representable lattices is one contribution of this paper. Another, perhaps more significant, contribution is the announcement of a novel approach to the discovery of new classes of representable lattices, the full potential of which we have only begun to explore.

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Correspondence to William DeMeo.

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Presented by J. Berman.

This article is dedicated to Ralph Freese and Bill Lampe.

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DeMeo, W. Expansions of finite algebras and their congruence lattices. Algebra Univers. 69, 257–278 (2013). https://doi.org/10.1007/s00012-013-0226-3

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  • DOI: https://doi.org/10.1007/s00012-013-0226-3

2010 Mathematics Subject Classification

  • Primary: 08A30
  • Secondary: 08A60
  • 06B10

Key words and phrases

  • congruence lattice
  • finite algebra
  • finite lattice representations