Journal Article: Expansions of Finite Algebras and Their Congruence Lattices
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README.md

Overalgebras

This repository contains a journal article as well as the GAP software that was used for constructing new finite algebras by extending and expanding transitive G-sets, as described in the paper.

Title: Expansions of finite algebras and their congruence lattices
Author: William DeMeo @williamdemeo
Journal: Algebra Universalis
Year: 2013

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 (B0,...), let B1, B2, ..., BK be sets that either intersect B0 or intersect each other at certain points. We construct an overalgebra (A,FA), by which we mean an expansion of (B0,...) with universe that is the union of the Bi's and a certain set FA of unary operations that includes mappings ei satisfying ei2 = ei and ei(A) = Bi, for 0 <= i <= K.

We explore two such constructions and prove results about the shape of the new congruence lattices Con(A,FA) 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.

BibTeX entry:

@article {DeMeo:2013,
    AUTHOR = {DeMeo, William},
     TITLE = {Expansions of finite algebras and their congruence lattices},
   JOURNAL = {Algebra Universalis},
  FJOURNAL = {Algebra Universalis},
    VOLUME = {69},
      YEAR = {2013},
    NUMBER = {3},
     PAGES = {257--278},
      ISSN = {0002-5240},
   MRCLASS = {08A30 (06B10 08A60)},
  MRNUMBER = {3041715},
MRREVIEWER = {E. W. Kiss},
       DOI = {10.1007/s00012-013-0226-3},
       URL = {http://dx.doi.org/10.1007/s00012-013-0226-3},
}

The published version of the paper is in the file DeMeo-Expansions-AU-2013.pdf, and is also available at springer.com.

In the file gap2uacalc.g is a GAP program that can be used on its own to convert GAP groups and G-sets into UACalc .ua files, which can then be imported into the Universal Algebra Calculator. See universalalgebra.org for more information about gap2uacalc.g.

For questions, comments, or suggestions please submit an issue.

Thanks for your interest in this work!

@williamdemeo