Distributivity in Quandles and Quasigroups

Abstract

Distributivity in algebraic structures appeared in many contexts such as in quasigroup theory, semigroup theory and algebraic knot theory. In this paper we give a survey of distributivity in quasigroup theory and in quandle theory

Latin quandles and applications to cryptography

“…x = id for all x ∈ X [10]. Joyce reported a class of quandles in which the symmetries S(y) are all involutions [49].…”

“…However, the notion of self-distributive binary algebra is not new in literature. It appears with many different names [10,16]. One of the earliest examples is the work of Burstin and Mayer of 1929 [4,49].…”

“…The foregoing shows that there are many examples of quandles. For a detailed study of variuos examples of quandle structures, the reader can check the following references [1,2,8,[10][11][12][13]15,34,35]. Quandles provide several invariants of knots, especially the class of connected quandles.…”

Computing with Colored Tangles

“…The first person to consider distributivity as an axiom of primary importance, and to suggest diagrammatic calculi for logic (and perhaps by extension to computation) was American philosopher Charles Saunders Peirce. The following Peirce quotation was pointed out by Elhamdadi [19]: "These are other cases of the distributive principle. .…”

Dynamical Cocycles Associated With Certain Non-Medial Left-Distributive Quasigroups

“…Finally Vendramin [14] and Vlachy [15] 52 HYUN-JONG SONG AND HYO-SEOB SIM independently proved that there are no more non-medial left-distributive quasigroups of order 15 except for G[Z 5 , [0]) and G[Z 5 , [1]). In his survey article [4] Elhamdadi reviewed on the progress mentioned in the above. The purpose of this paper is to explicitly present Andruskiewitsch-Graña's dynamical cocycles [1] associated with G 15 = G(Z 5 , [0]) and S 15 = G(Z 5 , [1]), which are extensions of the dihedral quandle of order 3 (referred to as the Tait quandle for short) by those cocycles.…”