Hoops

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

Subclasses

Superclasses