ortholattices - MathStructures

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=====Ortholattices=====

Abbreviation: **OLat**

====Definition====
An \emph{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:  $x\vee x'=1$, $x\wedge x'=0$, $x''=x$

$'$ 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====
^[[Classtype]]  |variety |
^[[Equational theory]]  |decidable |
^[[Quasiequational theory]]  | |
^[[First-order theory]]  |undecidable |
^[[Locally finite]]  |no |
^[[Residual size]]  |unbounded |
^[[Congruence distributive]]  |yes |
^[[Congruence modular]]  |yes |
^[[Congruence n-permutable]]  | |
^[[Congruence regular]]  | |
^[[Congruence uniform]]  | |
^[[Congruence extension property]]  |no |
^[[Definable principal congruences]]  |no |
^[[Equationally def. pr. cong.]]  |no |
^[[Amalgamation property]]  | Yes |
^[[Strong amalgamation property]]  | Yes [(BrunsHarding1997)] |
^[[Epimorphisms are surjective]]  | |

====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 | 60 |  0 | 311 |  0 |    |  0 |    |  0 |    |  0 |    |  0 |    |  0 |
^# of si's | 0 | 1 | 0 | 0 | 0 | 2 | 0 | 3 | 0 | 11 |  0 | 45 |  0 | 240 |  0 |    |  0 |    |  0 |    |  0 |    |  0 |    |  0 |

====Subclasses====
[[Orthomodular lattices]] 


====Superclasses====
[[Complemented lattices]] 


====References====

[(BrunsHarding1997>
G. Bruns and J. Harding, \emph{Amalgamation of ortholattices}, Order 14 (1997/98), no. 3, 193–209
[[http://www.ams.org/mathscinet-getitem?mr=99f:06014|MRreview]])]

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