A type of algebraic structure related to sets of intervals

TL;DR

This paper investigates conditions under which a family of subsets of a set can be ordered so that all are convex, characterizing the structure of such families and connecting to interval graphs and hypergraphs.

Contribution

It provides necessary and sufficient conditions for finite families of subsets to be convex in some total order, extending to infinite cases and establishing bounds on related set structures.

Findings

Characterization of the closure of subset families under convex operations

Necessary and sufficient conditions for ordering subsets to be convex

Bounds on the size of generated set structures

Abstract

F. Wehrung has asked: Given a family $\mathcal{C}$ of subsets of a set $\Omega$, under what conditions will there exist a total ordering on $\Omega$ under which every member of $\mathcal{C}$ is convex? <p> Note that if $A$ and $B$ are nondisjoint convex subsets of a totally ordered set, neither of which contains the other, then $A\cup B$, $A\cap B$, and $A\setminus B$ are also convex. So let $\mathcal{C}$ be an arbitrary set of subsets of a set $\Omega$, and form its closure $\mathcal{P}$ under forming, whenever $A$ and $B$ are nondisjoint and neither contains the other, the sets $A\cup B$, $A\cap B$, and $A\setminus B$. We determine the form $\mathcal{P}$ can take when $\mathcal{C}$, and hence $\mathcal{P}$, is finite, and for this case get necessary and sufficient conditions for there to exist an ordering of $\Omega$ of the desired sort. From this we obtain a condition which works without the finiteness hypothesis. <p> We establish bounds on the cardinality of the subset $\mathcal{P}$ generated as above by an $n$-element set $\mathcal{C}$. <p> We note connections with the theory of <i>interval graphs</i> and <i>hypergraphs</i>, which lead to other ways of answering Wehrung's question.