Table of Contents
Complemented modular lattices
Abbreviation: CdMLat
Definition
A \emph{complemented modular lattice} is a complemented lattices $\mathbf{L}=\langle L,\vee ,0,\wedge ,1\rangle $ that is
modular lattices: $(( x\wedge z) \vee y) \wedge z=( x\wedge z) \vee ( y\wedge z) $
Morphisms
Let $\mathbf{L}$ and $\mathbf{M}$ be complemented modular lattices. A morphism from $\mathbf{L}$ to $\mathbf{M}$ is a function $h:L\to M$ 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:
Basic results
This class generates the same variety as the class of its finite members plus the non-desargean planes.
Properties
| Classtype | first-order |
|---|---|
| Equational theory | decidable |
| Quasiequational theory | undecidable |
| 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 | |
| Definable principal congruences | |
| Equationally def. pr. cong. | |
| Amalgamation property | |
| Strong amalgamation property | |
| Epimorphisms are surjective |
Finite members
$\begin{array}{lr}
f(1)= &1
f(2)= &1
f(3)= &0
f(4)= &1
f(5)= &1
f(6)= &
f(7)= &
f(8)= &
\end{array}$
