# Directoids

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### 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.

### 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

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Semilattices

### Superclasses

Groupoids

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