Congruence lattices of algebras---

the signed labelling

Author:
S. W. Seif

Journal:
Proc. Amer. Math. Soc. **124** (1996), 1361-1370

MSC (1991):
Primary 08A30

DOI:
https://doi.org/10.1090/S0002-9939-96-03102-4

MathSciNet review:
1301526

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Abstract: For an arbitrary algebra a new labelling, called the signed labelling, of the Hasse diagram of is described. Under the signed labelling, each edge of the Hasse diagram of receives a label from the set . The signed labelling depends completely on a subset of the unary polynomials of 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 admits a natural join congruence, denoted , such that satisfies the semimodular law. In an application of that result, it is shown that for a regular semigroup , for which in , is actually a lattice congruence, coincides with , and satisfies the semimodular law.

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Additional Information

**S. W. Seif**

Affiliation:
Department of Mathematics, University of Louisville, Louisville, Kentucky 40292

Email:
swseif01@homer.louisville.edu

DOI:
https://doi.org/10.1090/S0002-9939-96-03102-4

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