Semilattices with zero

A semilattice with zero is a structure S = (S,·,0) of type (2,0) such that
(S,·) is a semilattice and
0 is a zero for ·: x·0 = 0.

Let S and T be semilattices with zero. A morphism from S to T is a function h : S→T that is a homomorphism: h(x·y) = h(x)·h(y) and h(0) = 0.

Classtype	variety
Equational theory	decidable in PTIME
Quasiequational theory	decidable
First-order theory	undecidable
Locally finite	no
Residual size	unbounded
Congruence distributive	no
Congruence modular	no
Congruence n-permutable	no
Congruence regular	no
Congruence uniform	no
Congruence extension property
Definable principal congruences
Equationally definable principal congruences
Amalgamation property
Strong amalgamation property
Epimorphisms are surjective

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

[Semilattices with identity and zero]?