homogeneous group

A homogeneous group is a set $G$ together with a map $():G\times G\times G\to G$ satisfying:

i)$(a,a,b)=b$

ii)$(a,b,b)=a$

iii)$((a,b,c),d,e)=(a,b,(c,d,e))$

for all $a,b,c,d,e\in G$.

A map $f:G\to H$ of homogeneous groups is a homomorphism if it $f(a,b,c)=(fa,fb,fc)$, for all $a,b,c\in G$.

A non-empty homogeneous group is essentially a group, as given any $x\in G$, we may define the following product on $G$:

$ab=(a,x,b)$.

This gives $G$ the of a group with identity $x$. The choice of $x$ does not affect the isomorphism class of the group obtained.

One may recover a homogeneous group from a group obtained this way, by setting

$(a,b,c)=a{b}^{-1}c$.

Also, every group may be obtained from a homogeneous group.

Homogeneous groups are homogeneous: Given $a,b\in G$ we have a homomorphism $f$ taking $a$ to $b$, given by $fx=(x,a,b)$.

In this way homogeneous groups differ from groups, as whilst often used to describe symmetry, groups themselves have a distinct element: the identity.

Also the definition of homogeneous groups is given purely in of identities, and does not exclude the empty set.