−Table of Contents
Semilattices with zero
Abbreviation: Slat0
Definition
A \emph{semilattice with zero} is a structure S=⟨S,⋅,0⟩ of type ⟨2,0⟩ such that
⟨S,⋅⟩ is a semilattices
0 is a zero for ⋅: x⋅0=0
Morphisms
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), h(0)=0
Examples
Example 1:
Basic results
Properties
Classtype | variety |
---|---|
Equational theory | decidable in PTIME |
Quasiequational theory | decidable |
First-order theory | undecidable |
Locally finite | yes |
Residual size | 2 |
Congruence distributive | no |
Congruence modular | no |
Congruence meet-semidistributive | yes |
Congruence n-permutable | no |
Congruence regular | no |
Congruence uniform | no |
Congruence extension property | yes |
Definable principal congruences | yes |
Equationally def. pr. cong. | no |
Amalgamation property | yes |
Strong amalgamation property | yes |
Epimorphisms are surjective | yes |
Finite members
f(1)=1f(2)=1f(3)=2f(4)=5f(5)=15f(6)=53