Abbreviation: Dtoid

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

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

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

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

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

Remark:

Let $\mathbf{A}$ and $\mathbf{B}$ be directoids. 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:

The relation $x\le y \iff x\cdot y=x$ is a partial order.

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

Semilattices

Groupoids