Some Properties of Inclusions of Multisets and Contractive Boolean Operators

TL;DR

This paper explores the properties of a set inclusion preorder based on unordered sections, characterizes it through boolean operators, and discusses the complexity of generating these operators.

Contribution

It provides a clear description of the equivalence classes and the preorder, and analyzes the non-finite generation of relevant boolean operators.

Findings

Equivalence classes are easily characterized.

Preorder can be described via pointwise inclusion and boolean operators.

Relevant operators are not finitely generated.

Abstract

Consider the following curious puzzle: call an n-tuple X=(X_1, ..., X_n) of sets smaller than another n-tuple Y if it has fewer //unordered sections//. We show that equivalence classes for this preorder are very easy to describe and characterize the preorder in terms of the simpler pointwise inclusion and the existence of a special increasing boolean operator f:B^n -> B^n. We also show that contrary to increasing boolean operators, the relevant operators are not finitely generated, which might explain why this preorder is not easy to describe concretely.