Subgroups of $SF(\omega)$ and the relation of almost containedness

TL;DR

This paper explores the structure of the lattice of finitary permutation groups, defining new cardinal invariants related to almost containedness and orthogonality, and investigates their consistency with Ramsey-type theorems.

Contribution

It introduces six cardinal numbers associated with these relations and analyzes their consistency and connections to Ramsey theory.

Findings

Defined six cardinal invariants for the relations

Established some consistency results for these invariants

Linked the invariants to variants of Ramsey's theorem

Abstract

The relations of almost containedness and orthogonality in the lattice of groups of finitary permutations are studied in the paper. We define six cardinal numbers naturally corresponding to to these relations by the standard scheme of $P(\omega)$. We obtain some consistency results concerning these numbers and some versions of the Ramsey theorem.