Double Magma associated with Ward and double Ward quasigroups

TL;DR

This paper explores the structure of double magmas related to Ward quasigroups and their duals, establishing conditions for their formation and revealing deep links between mediality and the interchange law in quasigroups.

Contribution

It introduces new conditions for forming double magmas from right modular, left unital magmas and clarifies the relationship between mediality and the interchange law in quasigroups.

Findings

Ward quasigroup double magma is improper

Conditions for forming double magma from certain magmas are established

Mediality is equivalent to the interchange law in quasigroups

Abstract

We describe types of double magma associated with Ward quasigroups, double Ward quasigroups, their duals and the groups they generate. Ward quasigroup double magma and unipotent, right modular, left unital double magma are proved to be improper. Necessary and sufficient conditions are found on a pair of right modular, left unital magma (and right-left unital magma) for them to form a double magma. We give further insight into the intimate connection between mediality and the interchange law by proving that a quasigroup is medial if and only if any pair of its parastrophic binary operations satisfy the interchange law.