# D-sets in Arbitrary Semigroup

## TL;DR

This paper introduces the concept of D-sets in arbitrary semigroups, explores their properties, and establishes their behavior under Cartesian products and in relation to C-sets in commutative cases.

## Contribution

It defines D-sets in general semigroups and proves their stability under Cartesian products and their relation to C-sets in commutative semigroups.

## Key findings

- Cartesian product of finitely many D-sets is a D-set
- Partial results for infinite Cartesian products of D-sets
- D-sets in commutative semigroups are C-sets

## Abstract

We define the notion of $D$-set in an arbitrary semigroup, and with some mild restrictions we establish its dynamical and combinatorial characterizations. Assuming a weak form of cancellation in semigroups we have shown that the Cartesian product of finitely many $D$-sets is a $D$-set. A similar partial result has been proved for Cartesian product of infinitely many $D$-sets. Finally, in a commutative semigroup we deduce that $D$-sets (with respect to a F{\o}lner net) are $C$-sets.