Document Zbl 1359.17008 - zbMATH Open

The following generalizations of objects with a binary operations are introduced:
1. A ternary shelf is a set \(Q\) with a ternary operation \(T: Q^3\longrightarrow T\) such that: \[ \forall x,y,z,u,v\in Q \:T(T(x,y,z),u,v)=T(T(x,u,v),T(y,u,v),T(z,u,v)). \]
2. A ternary rack is a ternary shelf \(Q\) such that for all \(y,z\in Q\), the map \(x\longrightarrow T(x,y,z)\) is invertible.
3. A ternary quandle is a ternary rack \(Q\) such that for all \(x\in Q\), \(T(x,x,x)=x\).
For examples, groups and binary quandles give rise to ternary quandles. A classification of quandles of cardinality 2 (2, up to an isomorphism) and of cardinality 3 (31, up to an isomorphism). Examples of ternary quandles shelves from groups, ternary bialgebras and 3-Lie algebras are described. Finally, a Hochschild-like theory of cohomology for ternary distributive algebras is introduced, and proved to fit with a theory of formal deformation of these objects.