Congruence lattices of algebras— the signed labelling

Abstract

For an arbitrary algebra A \mathbf {A} a new labelling, called the signed labelling, of the Hasse diagram of Con ⁡ A \operatorname {Con}\mathbf {A} is described. Under the signed labelling, each edge of the Hasse diagram of Con ⁡ A \operatorname {Con}\mathbf {A} receives a label from the set { + , − } \{+,-\} . The signed labelling depends completely on a subset of the unary polynomials of A \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 A \mathbf {A} admits a natural join congruence, denoted ≈ + \approx _+ , such that Con ⁡ A / ≈ + \operatorname {Con} \mathbf {A}/\approx _+ satisfies the semimodular law. In an application of that result, it is shown that for a regular semigroup S \mathbf {S} , for which J = D \mathcal {J}=\mathcal {D} in H ( S ) \mathbf {H}(\mathbf {S}) , ≈ + \approx _+ is actually a lattice congruence, ≈ + \approx _+ coincides with U U , and Con ⁡ S / U \operatorname {Con} \mathbf {S}/U ( = Con ⁡ S / ≈ + ) (=\operatorname {Con}\mathbf {S}/\approx _+) satisfies the semimodular law.