Medial groupoids
Definition
A medial groupoid is a structure $\mathbf{G}=\langle G,\cdot\rangle$, where $\cdot $ is an infix binary operation such that
$\cdot$ mediates: $(x\cdot y)\cdot(z\cdot w)=(x\cdot z)\cdot (y\cdot w)$
Morphisms
Let $\mathbf{G}$ and $\mathbf{H}$ be medial groupoids. A morphism from $\mathbf{G}$ to $\mathbf{H}$ is a function $h:G\rightarrow H$ that is a homomorphism:
$h(xy)=h(x)h(y)$
Jaroslav Jezek, Tomas Kepka,Equational theories of medial groupoids, Algebra Universalis, 171983,174–190MRreview
Jaroslav Jezek, Tomas Kepka,Medial groupoids, Rozpravy Ceskoslovenske Akad. Ved Rada Mat. Prirod. Ved, 931983,93MRreview
Examples
Example 1: $\langle S,*\rangle$, where $\langle S,+,\cdot\rangle$ is any commutative semiring, $a,b\in S$, and $x*y=a\cdot x+b\cdot y$.
Basic results
Properties
Finite members
$\begin{array}{lr} f(1)= &1\\ f(2)= &\\ f(3)= &\\ f(4)= &\\ f(5)= &\\ f(6)= &\\ f(7)= &\\ \end{array}$
Subclasses
Superclasses
References
Trace: » medial_groupoids