Table of Contents
Complemented lattices
Abbreviation: CdLat
Definition
A \emph{complemented lattice} is a bounded lattices $\mathbf{L}=\langle L,\vee ,0,\wedge ,1\rangle $ such that
every element has a complement: $\exists y(x\vee y=1\mbox{ and }x\wedge y=0)$
Morphisms
Let $\mathbf{L}$ and $\mathbf{M}$ be complemented lattices. A morphism from $\mathbf{L}$ to $\mathbf{M}$ is a function $h:L\rightarrow M$ that is a bounded lattice homomorphism:
$h(x\vee y)=h(x)\vee h(y)$, $h(x\wedge y)=h(x)\wedge h(y)$, $h(0)=0 $, $h(1)=1$
Examples
Example 1: $\langle P(S), \cup, \emptyset, \cap, S\rangle $, the collection of subsets of a set $S$, with union, empty set, intersection, and the whole set $S$.
Basic results
Properties
| Classtype | first-order |
|---|---|
| Equational theory | decidable |
| Quasiequational theory | |
| First-order theory | undecidable |
| Locally finite | no |
| Residual size | unbounded |
| Congruence distributive | yes |
| Congruence modular | yes |
| Congruence n-permutable | yes |
| Congruence regular | no |
| Congruence uniform | no |
| Congruence extension property | no |
| Definable principal congruences | no |
| Equationally def. pr. cong. | no |
| Amalgamation property | |
| Strong amalgamation property | |
| Epimorphisms are surjective |
Finite members
$\begin{array}{lr}
f(1)= &1
f(2)= &1
f(3)= &0
f(4)= &1
f(5)= &2
f(6)= &
f(7)= &
f(8)= &
\end{array}$
