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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Congruence lattices of algebras---
the signed labelling

Author: S. W. Seif
Journal: Proc. Amer. Math. Soc. 124 (1996), 1361-1370
MSC (1991): Primary 08A30
MathSciNet review: 1301526
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Abstract: For an arbitrary algebra $\mathbf{A}$ a new labelling, called the signed labelling, of the Hasse diagram of $\CON\mathbf{A}$ is described. Under the signed labelling, each edge of the Hasse diagram of $\CON\mathbf{A}$ receives a label from the set $\{+,-\}$. The signed labelling depends completely on a subset of the unary polynomials of $\mathbf{A}$ and its inspiration comes from semigroup theory. For finite algebras, the signed labelling complements the labelled congruence lattices of tame congruence theory (TCT). It provides a different kind of information about those algebras than the TCT labelling particularly with regard to congruence semimodularity. The main result of this paper shows that the congruence lattice of any algebra $\mathbf{A}$ admits a natural join congruence, denoted $\approx _+$, such that $\CON \mathbf{A}/\approx_+$ satisfies the semimodular law. In an application of that result, it is shown that for a regular semigroup $\mathbf{S}$, for which $\mathcal{J}=\mathcal{D}$ in $\mathbf{H}(\mathbf{S})$, $\approx_+$ is actually a lattice congruence, $\approx_+$ coincides with $U$, and $\CON \mathbf{S}/U$ $(=\CON\mathbf{S}/\approx_+)$ satisfies the semimodular law.

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S. W. Seif
Affiliation: Department of Mathematics, University of Louisville, Louisville, Kentucky 40292

Keywords: Congruence lattice, semimodularity
Received by editor(s): September 21, 1993
Received by editor(s) in revised form: September 7, 1994
Communicated by: Lance W. Small
Article copyright: © Copyright 1996 American Mathematical Society