Table of Contents
Semilattices with identity
Abbreviation: Slat$_1$
Definition
A semilattice with identity is a structure $\mathbf{S}=\langle S,\cdot,1\rangle$ of type $\langle 2,0\rangle $ such that
$\langle S,\cdot\rangle$ is a semilattices
$1$ is an indentity for $\cdot$: $x\cdot 1=x$
Morphisms
Let $\mathbf{S}$ and $\mathbf{T}$ be semilattices with identity. A morphism from $\mathbf{S}$ to $\mathbf{T}$ is a function $h:S\rightarrow T$ that is a homomorphism:
$h(x\cdot y)=h(x)\cdot h(y)$, $h(1)=1$
Examples
Example 1:
Basic results
Properties
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 def. pr. cong. | |
Amalgamation property | |
Strong amalgamation property | |
Epimorphisms are surjective |
Finite members
$\begin{array}{lr} f(1)= &1\\ f(2)= &\\ f(3)= &\\ f(4)= &\\ f(5)= &\\ f(6)= &\\ \end{array}$
Subclasses
Superclasses
References
Trace: » semilattices_with_identity