Baer sums for a natural class of monoid extensions

TL;DR

This paper extends the cohomological classification of group extensions to a broader class of monoid extensions called cosetal extensions, introducing a new Baer sum and a cohomology theory for them.

Contribution

It generalizes the second cohomology classification from groups to cosetal monoid extensions, defining a new Baer sum and associated cohomology.

Findings

Cosetal extensions are characterized by a new cohomology group.

A unique association between cosetal extensions and weakly Schreier split extensions is established.

A Baer sum operation for cosetal extensions is introduced.

Abstract

It is well known that the set of isomorphism classes of extensions of groups with abelian kernel is characterized by the second cohomology group. In this paper we generalise this characterization of extensions to a natural class of extensions of monoids, the cosetal extensions. An extension k: N -> G, e: G -> H, where k is the inclusion and e is the quotient , is cosetal if for all g,g' in G in which e(g) = e(g'), there exists a (not necessarily unique) n in N such that g = k(n)g'. These extensions generalise the notion of special Schreier extensions, which are themselves examples of Schreier extensions. Just as in the group case where a semidirect product could be associated to each extension with abelian kernel, we show that to each cosetal extension (with abelian group) kernel, we can uniquely associate a weakly Schreier split extension. The characterization of weakly Schreier split extensions is combined with a suitable notion of a factor set to provide a cohomology group granting a full characterization of cosetal extensions, as well as supplying a Baer sum.