## 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,

**17**1983,174–190MRreview

Jaroslav Jezek, Tomas Kepka,** Medial groupoids**,
Rozpravy Ceskoslovenske Akad. Ved Rada Mat. Prirod. Ved,

**93**1983,93MRreview

### 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