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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*·*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*(*x**y*) = *h*(*x*)*h*(*y*).

### Some results

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

### Examples

### Properties

### Finite members

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### Subclasses

Semilattices

### Superclasses

Groupoids