Ortholattices

Abbreviation: OLat

Definition

An ortholattice is a structure $\mathbf{L}=\langle L,\vee,0,\wedge,1,'\rangle$ such that

$\langle L,\vee,0,\wedge,1\rangle$ is a bounded lattice

$'$ is complementation: $x\vee x'=1$, $x\wedge x'=0$

$'$ satisfies De Morgan's laws: $(x\vee y)'=x'\wedge y'$, $(x\wedge y)'=x'\vee y'$

Morphisms

Let $\mathbf{L}$ and $\mathbf{M}$ be ortholattices. A morphism from $\mathbf{L}$ to $\mathbf{M}$ is a function $h:L\to M$ that is a homomorphism:

$h(x\vee y)=h(x)\vee h(y)$, $h(x\wedge y)=h(x)\wedge h(y)$, $h(x')=h(x)'$

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

Finite members

$n$ 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25
# of algs 1 1 0 1 0 2 0 5 0 15 0 0 0 0 0 0 0 0
# of si's 0 1 0 0 0 2 0 3 0 11 0 0 0 0 0 0 0 0

Subclasses

Superclasses

References


1) G. Bruns and J. Harding, Amalgamation of ortholattices, Order 14 (1997/98), no. 3, 193–209 MRreview