, Volume 8, Issue 1, pp 35-65
Date: 11 Aug 2012

Homology of distributive lattices

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Abstract

We outline the theory of sets with distributive operations: multishelves and multispindles, with examples provided by semi-lattices, lattices and skew lattices. For every such a structure we define multi-term distributive homology and show some of its properties. The main result is a complete formula for the homology of a finite distributive lattice. We also indicate the answer for unital spindles and conjecture the general formula for semi-lattices and some skew lattices. Then we propose a generalization of a lattice as a set with a number of idempotent operations satisfying the absorption law.

Communicated by Jim Stasheff.
JHP was partially supported by the NSA-AMS 091111 Grant, by the Polish Scientific Grant: Nr. N-N201387034, and by the GWU REF Grant. KKP was supported by the NSF Grant DMS-1005750 in summer 2011.
The paper is dedicated to Paweł Waszkiewicz (1973–2011), who was a faculty member of Theoretical Computer Science at Jagiellonian University in Krakow. He obtained PhD at the University of Birmingham, UK, in 2002 in the theory of domains and formal languages. Although his career has been ceased in a tragic manner 9 years later, he had already published 21 papers. The second author is indebted to him for being introduced to the fascinating world of categories, posets and domains.