Doubling Tolerances and Coalition Lattices
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 doubling tolerance on 𝐿. We prove that, in this case, 𝐿 and 𝑇 determine a modular lattice of size 2|𝐿|. This construction preserves distributivity. In order to give an application of the new construct, let 𝑃 be a partially ordered set (poset). Following a 1995 paper by G. Pollák and the present author, the subsets of 𝑃 are called the coalitions of 𝑃. For coalitions X and 𝑌 of 𝑃, let 𝑋 ≤ 𝑌 mean that there exists an injective map 𝑓 from 𝑋 to 𝑌 such that 𝑥 ≤ 𝑓 (𝑥) for every 𝑥 ∈ 𝑋. If 𝑃 is a finite chain, then its coalitions form a distributive lattice by the 1995 paper; we give a new proof of the distributivity of this lattice by means of doubling tolerances.
Keywords: Lattice tolerance, modular lattice, coalition lattice
1991 Mathematics Subject Classification. 006B99, 06C99, 06D99