Abbreviation: OrdA

An order algebra is a structure $\mathbf{A}=\langle A,\cdot \rangle $, where $\cdot $ is an infix binary operation such that

$\cdot $ is idempotent: $x\cdot x=x$

$(x\cdot y)\cdot x=y\cdot x$

$(x\cdot y)\cdot y=x\cdot y$

$x\cdot ((x\cdot y)\cdot z)=x\cdot(y\cdot z)$

$((x\cdot y)\cdot z)\cdot y=(x\cdot z)\cdot y$

Remark:

Let $\mathbf{A}$ and $\mathbf{B}$ be order algebras. A morphism from $\mathbf{A}$ to $\mathbf{B}$ is a function $h:A\rightarrow B$ that is a homomorphism:

$h(xy)=h(x)h(y)$

Example 1:

$\begin{array}{lr} f(1)= &1\\ f(2)= &\\ f(3)= &\\ f(4)= &\\ f(5)= &\\ f(6)= &\\ f(7)= &\\ \end{array}$

Bands

Groupoids