Rivisiting the Hales--Jewett Theorem

TL;DR

This paper extends the Hales--Jewett theorem to semigroups with finite retraction families using ultrafilter tensor products.

Contribution

It introduces an abstract version of the Hales--Jewett theorem for semigroups with retractions, leveraging ultrafilter tensor products.

Findings

Established an abstract Hales--Jewett theorem for semigroups.

Utilized ultrafilter tensor products in the proof.

Provided a new perspective on combinatorial theorems in algebraic structures.

Abstract

This short note establishes an abstract Hales--Jewett theorem for semigroups equipped with a finite family of retractions. The proof relies on the interplay between retractions and tensor products of ultrafilters.