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### Definition

A *hoop* is a structure *A* = (*A*,·,→,1) of type (2,2,0) such that

(*A*,·,1) is a commutative monoid,

*x*→( *y*→*z*) = (*x*·*y*)→*z*,

*x*→*x* = 1, and

(*x*→*y*)·*x* = (*y*→*x*)·*y*.

**Remark**:
This definition shows that hoops form a variety.

Hoops are partially ordered by the relation *x* ≤ *y* ⇔
*x*→*y* = 1.

The operation *x*∧*y* = (*x*→*y*)·*x* is a meet with
respect to this order.

### Morphisms

Let *A* and *B* be hoops. A morphism from *A* to *B* is a function *h* : *A*→*B* that is a homomorphism:
*h*(*x*·*y*) = *h*(*x*)·*h*(*y*) and *h*(*x*→*y*) = *h*(*x*)→*h*(*y*) and *h*(1) = 1.

### Some results

### Examples

### Properties

### Finite members

[Size 1]?: 1

[Size 2]?: 1

[Size 3]?:

[Size 4]?:

[Size 5]?:

[Size 6]?:

[Size 7]?:

### Subclasses

Wajsberg hoops

[Idempotent hoops]?

[Commutative generalized BL-algebras]?

### Superclasses

Pocrims?

[Generalized hoops]?