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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 $h:Larrow M$ 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 $S$.

Properties

Classtype variety decidable undecidable no unbounded yes yes no no no

Finite members

$\begin{array}{lr} f(1)= &1\\ f(2)= &1\\ f(3)= &\\ f(4)= &1\\ f(5)= &\\ f(6)= &\\ f(7)= &\\ \end{array}$