A \emph{hoop} is a structure $\mathbf{A}=\langle A,\cdot,\rightarrow,1\rangle$ of type $\langle 2,2,0\rangle$ such that

$\langle A,\cdot ,1\rangle$ is a commutative monoid

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

$x\rightarrow x=1$

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

Remark: This definition shows that hoops form a variety.

Hoops are partially ordered by the relation $x\leq y \iff x\rightarrow y=1$.

The operation $x\wedge y = (x\rightarrow y)\cdot x$ is a meet with respect to this order.

A \emph{hoop} is a structure $\mathbf{A}=\langle A,\cdot,\rightarrow,1\rangle$ of type $\langle 2,2,0\rangle$ such that

$x\cdot y = y\cdot x$

$x\cdot 1 = x$

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

$x\rightarrow x=1$

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

A \emph{hoop} is a structure $\mathbf{A}=\langle A,\cdot,\rightarrow,1\rangle$ 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 $x\rightarrow y = 1$ then

$\le$ is a partial order,

$\rightarrow$ is the residual of $\cdot$, i.e., $\ x\cdot y\le z \iff y\le x\rightarrow z$, and

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

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:

$h(x\cdot y)=h(x)\cdot h(y)$, $h(x\rightarrow y)=h(x)\rightarrow h(y)$, $h(1)=1$

Example 1:

Finite hoops are the same as generalized BL-algebras (= divisible residuated lattices) since the join always exists in a finite meet-semilattice with top, and since all finite GBL-algebras are commutative and integral.

$\begin{array}{lr} f(1)= &1 f(2)= &1 f(3)= &2 f(4)= &5 f(5)= &10 f(6)= &23 f(7)= &49 \end{array}$

Wajsberg hoops

Idempotent hoops

Commutative generalized BL-algebras

Pocrims

Generalized hoops