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11 Aug 2012
Homology of distributive lattices
 Józef H. Przytycki,
 Krzysztof K. Putyra
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We outline the theory of sets with distributive operations: multishelves and multispindles, with examples provided by semilattices, lattices and skew lattices. For every such a structure we define multiterm 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 semilattices 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 NSAAMS 091111 Grant, by the Polish Scientific Grant: Nr. NN201387034, and by the GWU REF Grant. KKP was supported by the NSF Grant DMS1005750 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.
 Balbes, R., Dwinger, P. (1974) Distributive Lattices. University of Missouri Press, Columbia
 Bourbaki, N. (1989) Algebra I: Chapters 1–3, Elements of Mathematics. Springer, Berlin
 Carter, J.S.: A Survey of Quandle Ideas, the chapter in the book Introductory Lectures on Knot Theory: Selected Lectures presented at the Advanced School and Conference on Knot Theory and its Applications to Physics and Biology, ICTP, Trieste, Italy, 11–29 May 2009. World Scientific, Series on Knots and Everything, vol. 46, pp. 22–53 (2011)
 Carter, S., Jelsovsky, D., Kamada, S., Langford, L., Saito, M. (1999) Statesum invariants of knotted curves and surfaces from quandle cohomology. Electron. Res. Announc. Am. Math. Soc 5: pp. 146156 CrossRef
 Carter, S., Jelsovsky, D., Kamada, S., Saito, M. (2001) Quandle homology groups, their Betti numbers, and virtual knots. J. Pure Appl. Algebra 157: pp. 135155 CrossRef
 Carter, S., Kamada, S., Saito, M.: Surfaces in 4space. In: Gamkrelidze, R.V., Vassiliev, V.A. (eds.) Encyclopaedia of Mathematical Sciences, LowDimensional Topology III
 Clauwens, F.J.B.J. (2011) The algebra of rack and quandle cohomology. J. Knot Theory Ramif. 20: pp. 14871535 CrossRef
 Crans, A.S.: Lie 2algebras. PhD dissertation (2004). UC Riverside, arXiv:math.QA/0409602
 Etingof, P., Grana, M. (2003) On rack cohomology. J. Pure Appl. Algebra 177: pp. 4959 CrossRef
 Frabetti, A.: Dialgebra (co)homology with coefficients. In: Loday, J.L., Frabetti, A., Chapoton, F., Goichot, F. (eds.) Dialgebras and Related Operads. Lectures Notes in Mathematics, vol. 1763, pp. 67–103. Springer, Berlin (2001)
 Frabetti, A. (1997) Dialgebra homology of associative algebras. C.R. Acad. Sci. Paris 325: pp. 135140 CrossRef
 Fenn, R.: Tackling the Trefoils (preprint, 2011); arXiv:1110.0582v1 (will appear in the volume “Virtual knots” in JKTR)
 Fenn, R., Rourke, C. (1992) Racks and links in codimension two. J. Knot Theory Ramif. 1: pp. 343406 CrossRef
 Fenn, R., Rourke, C., Sanderson, B.J. (2004) James bundles and applications. Proc. Lond. Math. Soc. 3: pp. 217240 CrossRef
 Grätzer, G.: Lattice Theory. First concepts and distributive lattices. W.H. Freeman, San Francisco (1971) (Dover edition 2009)
 Greene, M.: Some results in geometric topology and geometry. PhD thesis, University of Warwick, advisor: Brian Sanderson (1997)
 Hochschild, G. (1945) On the cohomology groups of an associative algebra. Ann. Math. 46: pp. 5867 CrossRef
 Inasaridze, K.N. (1975) Homotopy of pseudosimplicial groups, nonabelian derived functors and algebraic Ktheory. Math. USSR Sbornik 98: pp. 339362
 Joyce, D. (1982) A classifying invariant of knots: the knot quandle. J. Pure Appl. Algebra 23: pp. 3765 CrossRef
 Leech, J.E. (1992) Normal skew lattices. Semigroup Forum 44: pp. 18 CrossRef
 Leech, J.E. (1996) Recent developments in the theory of skew lattices. Semigroup Forum 52: pp. 724 CrossRef
 Leech, J.E. (1989) Skew lattices in rings. A. Universalis 26: pp. 4872 CrossRef
 Litherland, R.A., Nelson, S. (2003) The Betti numbers of some finite racks. J. Pure Appl. Algebra 178: pp. 187202 CrossRef
 Loday, J.L.: Cyclic Homology. Grund. Math. Wissen. Band 301. Springer, Berlin (1992) (second edition, 1998)
 Niebrzydowski, M., Przytycki, J.H. (2006) Burnside Kei. Fundamenta Mathematicae 190: pp. 211229 CrossRef
 Niebrzydowski, M., Przytycki, J.H. (2009) Homology of dihedral quandles. J. Pure Appl. Algebra 213: pp. 742755 CrossRef
 Niebrzydowski, M., Przytycki, J.H. (2009) The quandle of the trefoil as the dehn quandle of the torus. Osaka J. Math. 46: pp. 645659
 Niebrzydowski, M., Przytycki, J.H. (2010) Homology operations on homology of quandles. J. Algebra 324: pp. 15291548 CrossRef
 Niebrzydowski, M., Przytycki, J.H. (2011) The second quandle homology of the Takasaki quandle of an odd abelian group is an exterior square of the group. J. Knot Theory Ramif. 20: pp. 171177 CrossRef
 T. Nosaka, On quandle homology groups of Alexander quandles of prime order. TAMS (submitted)
 Ohtsuki, T. (2003) Quandles, in Problems on invariants of knots and 3manifolds. Geom. Topol. Monogr. 4: pp. 455465
 Peirce, C.S. (1880) On the algebra of logic. Am. J. Math. 3: pp. 1557 CrossRef
 Przytycki, J.H. (2011) Distributivity versus associativity in the homology theory of algebraic structures. Demonstratio Math 44: pp. 823869
 Przytycki, J.H., Sikora, A.S.: Distributive products and their homology. Commun. Algebra (2012), arXiv:1105.3700v1
 Serre, J.P.: Lie algebras and Lie groups, lectures given at Harvard University (1964), 2nd edn. Lecture Notes in Mathematics, vol. 1500, Springer, Berlin (1992)
 Sikorski, R.: Boolean algebras. Springer, Berlin (1960) (second edition 1964)
 Takasaki, M.: Abstraction of symmetric transformation (in Japanese). Tohoku Math. J. 49, 145–207 (1942/1943); the English translation is being prepared by S. Kamada
 Tierney, M., Vogel, W. (1969) Simplicial derived functors in “Category theory, homology theory and applications”. Springer LNM 68: pp. 167179
 Traczyk, T.: Wstçp do teorii algebr Boole’a, Biblioteka Matematyczna, Tom 37, PWN, Warszawa (1970)
 Title
 Homology of distributive lattices
 Journal

Journal of Homotopy and Related Structures
Volume 8, Issue 1 , pp 3565
 Cover Date
 20130401
 DOI
 10.1007/s4006201200125
 Print ISSN
 21938407
 Online ISSN
 15122891
 Publisher
 SpringerVerlag
 Additional Links
 Topics
 Keywords

 Distributive homology
 Lattice
 Boolean algebra
 Spindle
 Multispindle
 Authors

 Józef H. Przytycki ^{(1)} ^{(2)}
 Krzysztof K. Putyra ^{(3)}
 Author Affiliations

 1. Department of Mathematics, George Washington University, Washington, DC, 20052, USA
 2. Institute of Mathematics, University of Gdańsk, Gdańsk, Poland
 3. Department of Mathematics, Columbia University, New York, NY, 10027, USA