Doubling tolerances and coalition lattices

Abstract

If every block of a (compatible) tolerance (relation) $T$ on a modular lattice $L$ of finite length consists of at most two elements, then we call $T$ a \emph{doubling tolerance} on $L$. We prove that, in this case, $L$ and $T$ determines a modular lattice of size $2|L|$. This construction preserves distributivity and modularity. In order to give an application of the new construct, let $P$ be a partially ordered set (poset). Following a 1995 paper by G.\ Poll\'ak and the present author, the subsets of $P$ are called the \emph{coalitions} of $P$. For coalitions $X$ and $Y$ of $P$, let $X\leq Y$ mean that there exists an injective map $f$ from $X$ to $Y$ such that $x\leq f(x)$ for every $x\in X$. If $P$ is a finite chain, then its coalitions form a distributive lattice by the 1995 paper; we give a new proof of its distributivity by means of doubling tolerances.