Ideals with Smital properties

TL;DR

This paper investigates the Smital Property for $\sigma$-ideals on Polish groups, exploring its variants, connections with chain conditions, maximality, and behavior under Fubini products.

Contribution

It introduces and analyzes several variants of the Smital Property, establishing their relationships with chain conditions, maximality, and product preservation in Polish groups.

Findings

Several variants of the Smital Property are characterized.

Connections between Smital properties and chain conditions are established.

Behavior of Smital properties under Fubini products is analyzed.

Abstract

A $\sigma$-ideal $\mathcal{I}$ on a Polish group $(X,+)$ has Smital Property if for every dense set $D$ and a Borel $\mathcal{I}$-positive set $B$ the algebraic sum $D+B$ is a complement of a set from $\mathcal{I}$. We consider several variants of this property and study their connections with countable chain condition, maximality and how well they are preserved via Fubini products.