4,474 research outputs found

    Knots and distributive homology: from arc colorings to Yang-Baxter homology

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    This paper is a sequel to my essay "Distributivity versus associativity in the homology theory of algebraic structures" Demonstratio Math., 44(4), 2011, 821-867 (arXiv:1109.4850 [math.GT]). We start from naive invariants of arc colorings and survey associative and distributive magmas and their homology with relation to knot theory. We outline potential relations to Khovanov homology and categorification, via Yang-Baxter operators. We use here the fact that Yang-Baxter equation can be thought of as a generalization of self-distributivity. We show how to define and visualize Yang-Baxter homology, in particular giving a simple description of homology of biquandles.Comment: 64 pages, 29 figures; to be published as a Chapter in: "New Ideas in Low Dimensional Topology", World Scientific, Vol. 5

    Generalized algebra within a nonextensive statistics

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    By considering generalized logarithm and exponential functions used in nonextensive statistics, the four usual algebraic operators : addition, subtraction, product and division, are generalized. The properties of the generalized operators are investigated. Some standard properties are preserved, e.g., associativity, commutativity and existence of neutral elements. On the contrary, the distributivity law and the opposite element is no more universal within the generalized algebra.Comment: 11 pages, no figure, TeX. Reports on Mathematical Physics (2003), in pres

    Trimness of Closed Intervals in Cambrian Semilattices

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    In this article, we give a short algebraic proof that all closed intervals in a γ\gamma-Cambrian semilattice Cγ\mathcal{C}_{\gamma} are trim for any Coxeter group WW and any Coxeter element γW\gamma\in W. This means that if such an interval has length kk, then there exists a maximal chain of length kk consisting of left-modular elements, and there are precisely kk join- and kk meet-irreducible elements in this interval. Consequently every graded interval in Cγ\mathcal{C}_{\gamma} is distributive. This problem was open for any Coxeter group that is not a Weyl group.Comment: Final version. The contents of this paper were formerly part of my now withdrawn submission arXiv:1312.4449. 12 pages, 3 figure

    Plurals, possibilities, and conjunctive disjunction

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