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.
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).
The relation x ≤ y ⇔ x·y = x is a partial order.
|Congruence types||semilattice (5)|
|Congruence extension property|
|Definable principal congruences|
|Equationally definable principal congruences||no|
|Strong amalgamation property|
|Epimorphisms are surjective|