On some combinatorial properties of the orbits on subsets

TL;DR

This paper introduces generalized orbit algebras to analyze how combinatorial properties characterize permutation group actions on subsets, highlighting differences from traditional orbit algebras.

Contribution

It presents a novel concept of generalized orbit algebras that are not derived from permutation groups, expanding the theoretical framework.

Findings

Generalized orbit algebras can characterize group actions on subsets.

The paper identifies properties distinguishing these algebras from classical orbit algebras.

A long-standing challenge in defining such algebras not arising from permutation groups is addressed.

Abstract

We introduce generalised orbit algebras. The purpose here is to measure how some combinatorial properties can characterize the action of a group of permutations on the subsets. The similarity with orbit algebras is such that it took the author a long time to find a generalised orbit algebra not arising from a permutation group.