A 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 monoids

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

$x\rightarrow x=1$

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

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.

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:

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

Wajsberg hoops

Idempotent hoops

Commutative generalized BL-algebras

Pocrims

Generalized hoops