Hoops

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Difference (from prior major revision) (no other diffs)

Changed: 41c41

 yes

 decidable

Changed: 57c57

 yes

Changed: 61c61

 yes

Definition

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.

Morphisms

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.

Properties

 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

[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]?

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