2019/09/20 by Czédli, Gábor
#06B99 #FOS: Mathematics #Rings and Algebras (math.RA)
paper · doi:10.48550/arxiv.1909.09539
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 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. 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 its distributivity by means of doubling tolerances.