Homology of Distributive Lattice

Abstract: While homology theory of associative structures, such as groups and rings, was extensively studied in the past, beginning with the work of Hopf, Eilenberg, and Hochschild, homology of non-associative distributive structures, such as quandles, has been neglected until recently. Distributive structures have been studied for a long time. In 1880, C.S. Peirce emphasized the importance of (right-) self-distributivity in algebraic structures. However, homology for these universal algebras was introduced only sixteen years ago by Fenn, Rourke, and Sanderson. We develop this theory in the historical context and propose a general framework to study homology of distributive structures. We illustrate the theory by computing some examples of 1-term and 2-term homology, and then by discussing 4-term homology for Boolean algebras and distributive lattices. We will start with a gentle introduction to distributive lattices and Boolean algebras (and their generalizations) for topologists, and with homology theory of distributive structures for logicians. We will end by outlining potential relations to Khovanov homology, via the Yang-Baxter operator.