[Home]Directoids

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Abbreviation: Dtoid

Definition

A directoid is a structure A = (A,·), where · is an infix binary operation such that
· is idempotent:   x·x = x
(x·yx = x·y
y·(x·y) = x·y
x·((x·yz) = (x·yz.

Remark:

Morphisms

Let A and B be directoids. A morphism from A to B is a function h : AB that is a homomorphism: h(xy) = h(x)h(y).

Some results

The relation x ≤ y   ⇔   x·y = x is a partial order.

Examples

Properties

Classtype variety
Equational theory
Quasiequational theory
First-order theory
Locally finite
residual size unbounded
Congruence distributive no
Congruence modular no
Congruence n-permutable no
Congruence regular no
Congruence uniform no
Congruence types semilattice (5)
Congruence extension property
Definable principal congruences
Equationally definable principal congruences no
Amalgamation property
Strong amalgamation property
Epimorphisms are surjective

Finite members

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

Subclasses

Semilattices

Superclasses

Groupoids


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Last edited April 19, 2003 10:07 pm (diff)
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