# Richness of arithmetic progression in commutative semigroup

## TL;DR

This paper extends the understanding of arithmetic progressions in piecewise syndetic sets within commutative semigroups, building on algebraic and combinatorial methods to generalize previous results.

## Contribution

It generalizes Beiglboeck's elementary proof to broader settings of commutative semigroups, enhancing the applicability of combinatorial techniques.

## Key findings

- Extended the elementary proof to new types of commutative semigroups
- Demonstrated the presence of arithmetic progressions in broader algebraic structures
- Provided a unified combinatorial framework for these results

## Abstract

Furstenberg and Glasner proved that for an arbitrary k in N, any piecewise syndetic set contains k term arithmetic progressions and such collection is also piecewise syndetic in Z: They used algebraic structure of beta N. The above result was extended for arbitrary semigroups by Bergelson and Hindman, again using the structure of Stone-Cech compactification of general semigroup. Beiglboeck provided an elementary proof of the above result and asked whether the combinatorial argument in his proof can be enhanced in a way which makes it applicable to a more abstract setting. In a recent work the second author of this paper and S.Jana provided an affirmative answer to Beiglboeck's question for countable commutative semigroup. In this work we will extend the result of Beiglboeck in different type of settings.