Weakly partitive families on infinite sets

Abstract

Given a finite or infinite set $S$ and a positive integer $k$, a {\em binary structure} $B$ of base $S$ and of rank $k$ is a function $(S\times S)\setminus\{(x,x);~x\in S\}\longrightarrow\{0,\ldots,k-1\}$. A subset $X$ of $S$ is an interval of $B$ if for $a,b\in X$ and $x\in S\setminus X$, $B(a,x)=B(b,x)$ and $B(x,a)=B(x,b)$. The family of intervals of $B$ satisfies the following: $\emptyset$, $\underline{B}$ and $\{x\}$, where $x\in \underline{B}$, are intervals of $B$; for every family $\mathcal{F}$ of intervals of $B$, the intersection of all the elements of $\mathcal{F}$ is an interval of $B$; given intervals $X$ and $Y$ of $B$, if $X\cap Y\neq\emptyset$, then $X\cup Y$ is an interval of $B$; given intervals $X$ and $Y$ of $B$, if $X\setminus Y\neq\emptyset$, then $Y\setminus X$ is an interval of $B$; for every up-directed family $\mathcal{F}$ of intervals of $B$, the union of all the elements of $\mathcal{F}$ is an interval of $B$. Given a set $S$, a family of subsets of $S$ is weakly partitive if it satisfies the properties above. After suitably characterizing the elements of a weakly partitive family, we propose a new approach to establish the following \cite{I91}: Given a weakly partitive family $\mathcal{I}$ on a set $S$, there is a binary structure of base $S$ and of rank $\leq 3$ whose intervals are exactly the elements of $\mathcal{I}$.