%0 Journal Article
%T Weakly partitive families on infinite sets
%A Pierre Ille
%A Robert Woodrow
%J Contributions to Discrete Mathematics
%D 2009
%I University of Calgary
%X 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 $(S imes S)setminus{(x,x);~xin S}longrightarrow{0,ldots,k-1}$. A subset $X$ of $S$ is an interval of $B$ if for $a,bin X$ and $xin Ssetminus X$, $B(a,x)=B(b,x)$ and $B(x,a)=B(x,b)$. The family of intervals of $B$ satisfies the following: $emptyset$, $underline{B}$ and ${x}$, where $xin underline{B}$, 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 $Xcap Y eqemptyset$, then $Xcup Y$ is an interval of $B$; given intervals $X$ and $Y$ of $B$, if $Xsetminus Y eqemptyset$, then $Ysetminus X$ 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 $mathcal{I}$.
%U http://cdm.math.ucalgary.ca/cdm/index.php/cdm/article/view/141