# Cohomology monoids of monoids with coefficients in semimodules II

## TL;DR

This paper explores the relationships between different cohomology monoids of monoids with semimodule coefficients and their connection to monoid and group extension problems, advancing the theoretical framework.

## Contribution

It establishes links between old and new cohomology monoids and extension problems, extending previous work to include third cohomology monoids.

## Key findings

- Old and new second cohomology monoids classify Schreier extensions.
- Third cohomology monoid relates to specific group extension problems.
- The work generalizes cohomology theory for monoids with semimodule coefficients.

## Abstract

We relate the old and new cohomology monoids of an arbitrary monoid $M$ with coefficients in semimodules over $M$, introduced in the author's previous papers, to monoid and group extensions. More precisely, the old and new second cohomology monoids describe Schreier extensions of semimodules by monoids, and the new third cohomology monoid is related to a certain group extension problem.