Directoids

Abbreviation: Dtoid

Definition

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

Remark:

Morphisms

Let A and B be directoids. A morphism from A to B is a function h : A→B 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