The variety (of finite semigroups) DAb is the class of finite semigroups whose regular 𝒟-classes form abelian groups, which makes it both similar but also fundamentally different to the variety DA of finite semigroups whose regular 𝒟-classes form aperiodic semigroups. This latter variety has intensively been studied and admits a range of interesting characterizations. One of these characterizations is of a combinatorial nature and uses rankers. These are instructions of the form “go to the next a on the left/right” and can be evaluated over a word. They were introduced by Weis and Immerman (based on work by Schwentick, Thérien and Vollmer) to define certain congruences over finite words, which can be used to define a family of semigroups such that a semigroup (or monoid) is in DA if and only if it is a divisor of one of the semigroups in the family. The advantage of this approach is that algebraic questions can be reduced to combinatorial ones, which often results in decision algorithms. Therefore, similar characterizations are also interesting for other varieties and, in fact, some have been found for sub-varieties of DA. However, it seems that the approach has not been successfully applied to varieties outside of A – the variety of aperiodic semigroups.

In the talk, we will discuss possible congruences for a similar characterization of DAb. We will look at how these can be used as a tool when checking whether a given equation holds over all semigroups in DAb and we will discuss the techniques used to prove the characterization.