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Weakly partitive families on infinite sets

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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 $(S imes S)setminus{(x,x);~xin S}longrightarrow{0,ldots,k-1}$. 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 $mathcal{I}$.

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