Complemented lattices
Abbreviation: CdLat
Definition
A 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
$\begin{array}{lr} f(1)= &1\\ f(2)= &1\\ f(3)= &0\\ f(4)= &1\\ f(5)= &2\\ f(6)= &\\ f(7)= &\\ f(8)= &\\ \end{array}$
Subclasses
Superclasses
References
Trace: » complemented_lattices