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

Abstract

The relations of almost containedness and othogonality in the lattice of groups of finitary permutations are studied in the paper. We define six cardinal numbers naturally corresponding to these relations by the standard scheme of P (ω). We obtain some consistency results concerning these numbers and some versions of the Ramsey theorem.2010 Mathematics Subject Classification: 03E35, 03E02.

Finitary shadows of compact subgroups of $$S(\omega )$$

“…The main results of the paper concern the structure of shadows of compact subgroups with respect to orthogonality and almost containedness. In this way, we extend several results from [5] concerning chains, antichains and reaping families. We assume that the reader knows some basic set theory, [4], [9], for example Martin's Axiom.…”

“…Let LF be the lattice of all subgroups of the group SF (ω). In [5] the author studied some van Douwen invariants of LF . In the classical case these cardinals describe properties of the lattice of subsets of ω with respect to the relations of almost containedness and orthogonality associated with the ideal of finite sets (see [10]).…”

“…In the classical case these cardinals describe properties of the lattice of subsets of ω with respect to the relations of almost containedness and orthogonality associated with the ideal of finite sets (see [10]). In [5] this approach is applied to LF and the ideal IF of all finite subgroups. The appropriate invariants corresponding to a, h, p, t, r, s were introduced and studied.…”

“…The appropriate invariants corresponding to a, h, p, t, r, s were introduced and studied. The aim of the present paper is to apply the methods of [5] to the subsemilattice of shadows of compact subgroups of S(ω), i.e. groups of the form SF (ω) ∩ C, where C is a compact subgroup of S(ω).…”

“…The group G 1 is almost contained in G 2 (G 1 ≤ a G 2 ) if G 1 is a subgroup of a group finitely generated over G 2 by elements of SF (ω). These relations were introduced in [5]. The van Douwen invariants for LF mentioned above were defined with respect to them.…”