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