Functions with a maximal number of finite invariant or internally-1-quasi-invariant sets or supersets

TL;DR

This paper investigates functions on sets, especially natural numbers, that have maximal finite subsets or supersets with specific invariance properties related to group dynamics and quasi-invariance.

Contribution

It characterizes functions on sets and natural numbers with maximal finite internally-$k$-quasi-invariant subsets or supersets, extending group action concepts.

Findings

Identifies functions where every finite subset is internally-$k$-quasi-invariant.

Determines functions on $ $ with all finite intervals internally-$k$-quasi-invariant.

Classifies functions where finite subsets have finite internally-$k$-quasi-invariant supersets.

Abstract

A relaxation of the notion of invariant set, known as $k$-quasi-invariant set, has appeared several times in the literature in relation to group dynamics. The results obtained in this context depend on the fact that the dynamic is generated by a group. In our work, we consider the notions of invariant and 1-internally-quasi-invariant sets as applied to an action of a function $f$ on a set $I$. We answer several questions of the following type, where $k \in \{0,1\}$: what are the functions $f$ for which every finite subset of $I$ is internally-$k$-quasi-invariant? More restrictively, if $I = \mathbb{N}$, what are the functions $f$ for which every finite interval of $I$ is internally-$k$-quasi-invariant? Last, what are the functions $f$ for which every finite subset of $I$ admits a finite internally-$k$-quasi-invariant superset? This parallels a similar investigation undertaken by C. E. Praeger in the context of group actions.