# medial quasigroup

A medial quasigroup is a quasigroup such that, for any choice of four elements $a,b,c,d$, one has

 $(a\cdot b)\cdot(c\cdot d)=(a\cdot c)\cdot(b\cdot d).$

Any commutative quasigroup is trivially a medial quasigroup. A nontrivial class of examples may be constructed as follows. Take a commutative group $(G,+)$ and two automorphisms $f,g\colon G\to G$ which commute with each other, and an element $c$ of $G$. Then, if we define an operation $\cdot\colon G\times G\to G$ as

 $x\cdot y=f(a)+g(b)+c,$

$(G,\cdot)$ is a medial quasigroup.

Reference:

V. D. Belousov, Fundamentals of the theory of quasigroups and loops (in Russian)

Title medial quasigroup MedialQuasigroup 2013-03-22 16:27:33 2013-03-22 16:27:33 rspuzio (6075) rspuzio (6075) 5 rspuzio (6075) Definition msc 20N05