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.
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.
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.
|Congruence extension property|
|Definable principal congruences|
|Equationally definable principal congruences|
|Strong amalgamation property|
|Epimorphisms are surjective|