Balanced and functionally balanced $P$-groups

TL;DR

This paper investigates the conditions under which $P$-groups are balanced or functionally balanced, establishing equivalences and focusing on uniform free topological groups over uniform $P$-spaces.

Contribution

It proves that in $ ext{c}$-bounded $P$-groups, balancedness is equivalent to functional balancedness, and relates these properties to free topological groups over uniform $P$-spaces.

Findings

A $ ext{c}$-bounded $P$-group is balanced iff it is functionally balanced.

For any $P$-group, functional balancedness is equivalent to strong functional balancedness.

The balancedness of the uniform free topological group is characterized by the balancedness of subsets $B_n$.

Abstract

In relation to Itzkowitz's problem, we show that a $\mathfrak c$-bounded $P$-group is balanced if and only if it is functionally balanced. We prove that for an arbitrary $P$-group, being functionally balanced is equivalent to being strongly functionally balanced. A special focus is given to the uniform free topological group defined over a uniform $P$-space. In particular, we show that this group is (functionally) balanced precisely when its subsets $B_n,$ consisting of words of length at most $n,$ are all (resp., functionally) balanced.