A \emph{right hoop} is a structure $\mathbf{A}=\langle A,\cdot,/,1\rangle $ of type $\langle 2,2,0\rangle$ such that
$\langle A,\cdot ,1\rangle $ is a monoid
$x/(y\cdot z) = (x/z)/y$
$x/x=1$
$(x/y)\cdot y = (y/x)\cdot x$
Remark: This definition shows that right hoops form a variety.
Right hoops are partially ordered by the relation $x\leq y \iff y/x=1$.
The operation $x\wedge y = (x/y)\cdot y$ is a meet with respect to this order.
A \emph{right hoop} is a structure $\mathbf{A}=\langle A,\cdot,/,1\rangle $ of type $\langle 2,2,0\rangle$ such that
$x\cdot y = y\cdot x$
$x\cdot 1 = x$
$x/(y\cdot z) = (x/z)/y$
$x/x=1$
$(x/y)\cdot y = (y/x)\cdot x$
A \emph{right hoop} is a structure $\mathbf{A}=\langle A,\cdot,/,1\rangle $ of type $\langle 2,2,0\rangle$ such that
$\langle A,\cdot ,1\rangle $ is a commutative monoid
and if $x\le y$ is defined by $y/x = 1$ then
$\le$ is a partial order,
$/$ is the right residual of $\cdot$, i.e., $\ x\cdot y\le z \iff x\le z/y$, and
$(x/y)\cdot y = (y/x)\cdot x$.
Let $\mathbf{A}$ and $\mathbf{B}$ be hoops. A morphism from $\mathbf{A}$ to $\mathbf{B}$ is a function $h:A\rightarrow B$ that is a homomorphism:
$h(x\cdot y)=h(x)\cdot h(y)$, $h(x/y)=h(x)/h(y) $, $h(1)=1$
Example 1:
$\begin{array}{lr}
f(1)= &1
f(2)= &1
f(3)= &2
f(4)= &8
f(5)= &24
f(6)= &91
f(7)= &
\end{array}$