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:

$\begin{array}{lr} f(1)= &1 f(2)= &1 f(3)= &2 f(4)= &8 f(5)= &24 f(6)= &91 f(7)= & \end{array}$

hoops

Porrims