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A hoop is a structure A = (A,·,→,1) of type (2,2,0) such that
(A,·,1) is a commutative monoid,
x→( yz)  = (x·y)→z,
xx = 1, and
(xyx  = (yxy.

Remark: This definition shows that hoops form a variety.
Hoops are partially ordered by the relation x ≤ y   ⇔   xy = 1.
The operation xy  = (xyx is a meet with respect to this order.


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

Some results



Classtype variety
Equational theory decidable
Quasiequational theory decidable
First-order theory
Locally finite no
Residual size unbounded
Congruence distributive yes
Congruence modular yes
Congruence n-permutable
Congruence regular
Congruence uniform
Congruence extension property
Definable principal congruences
Equationally definable principal congruences
Amalgamation property
Strong amalgamation property
Epimorphisms are surjective

Finite members

[Size 1]?:  1
[Size 2]?:  1
[Size 3]?:  
[Size 4]?:  
[Size 5]?:  
[Size 6]?:  
[Size 7]?:  


Wajsberg hoops
[Idempotent hoops]?
[Commutative generalized BL-algebras]?


[Generalized hoops]?

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Last edited June 4, 2003 6:27 am (diff)