Linear diquasigroups

Abstract

Following the prototype of dimonoids, diquasigroups are directed versions of quasigroups, where the structure is split into left and right quasigroups on the same set. The linear and affine diquasigroups that form the topic of this paper are built on the foundation of a module. In this context, various issues that may be difficult to handle in the general case, for example identification of the largest two-sided quasigroup image, become more tractable. An appropriate universal algebraic language for affine diquasigroups is established, and the entropic models of this language are characterized. Various interesting classes of linear and affine diquasigroups are singled out for special attention, such as internally associative, Bol, and symmetric diquasigroups. The problem of determining which linear diquasigroups have an abelian group as their undirected replica is raised. One sufficient condition is provided, formulated in terms of a differential calculus for one-sided quasigroup words.