Doubling tolerances and coalition lattices

TL;DR

This paper introduces doubling tolerances on modular lattices, showing they produce larger lattices with preserved properties, and applies this to prove distributivity of coalition lattices in finite chains.

Contribution

It defines doubling tolerances on modular lattices, constructs larger lattices from them, and applies this to establish distributivity of coalition lattices in finite chains.

Findings

Doubling tolerances produce lattices of size twice the original.

Construction preserves modularity and distributivity.

Coalition lattices in finite chains are distributive.

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.