Ternary Distributive Structures and Quandles

Abstract

We introduce a notion of ternary distributive algebraic structure, give examples, and relate it to the notion of a quandle. Classification is given for low order structures of this type. Constructions of such structures from ternary bialgebras are provided. We also describe ternary distributive algebraic structures coming from groups and give examples from vector spaces whose bases are elements of a finite ternary distributive set. We introduce a cohomology theory that is analogous to Hochschild cohomology and relate it to a formal deformation theory of these structures.

Higher arity self-distributive operations in Cascades and their cohomology

“…The binary case has been studied extensively in the literature under the name of quandle, and its connections to low dimensional topology have been widely exploited, see for example [6,9]. The ternary self-distributivity and its cohomology theory that generalizes the binary case have been also studied [13,21]. In this paper we produce ternary operations from mutually distributive binary operations.…”

“…In particular, consider Z 8 with the ternary operation T (x, y, z) = 3x + 2y + 4z. This affine ternary rack given in [13] is not induced by an Alexander quandle structure as described in the preceding item since 3 is not a square in Z 8 .…”

Ternary and n-ary f-distributive structures

“…One may also mention reference [6] where the author uses two ternary operators, providing a generalization of a Dehn presentation which assigns a relation to each crossing in terms of the regions of the diagram that surround the crossing. For example, by coloring the four regions respectively a, b, c and d (see gure 2 in [6]), the author obtains d as a ternary function T(a, b, c) = ab − c. This example of ternary operation was also considered in Example 2.8 in [5]. The author shows under certain conditions that ternary checkerboard colorings de ne link invariants.…”

“…In Lie algebra theory, for example, the bracket is replaced by a n-ary bracket and the Jacobi identity is replaced by its higher analogue, see [4]. Generalizations of quandles to the ternary case were done recently in [5]. One may also mention reference [6] where the author uses two ternary operators, providing a generalization of a Dehn presentation which assigns a relation to each crossing in terms of the regions of the diagram that surround the crossing.…”

“…The author shows under certain conditions that ternary checkerboard colorings de ne link invariants. The paper deals with di erent algebraic structures considered in [5] and in this work. In this paper we introduce and study a twisted version of ternary, respectively n-ary, generalizations of racks and quandles, where the structure is de ned by a ternary operation and a linear map twisting the distributive property.…”