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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 $h:L\to M$ 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 ^{1)} |

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 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | |||||||

# of si's | 0 | 1 | 0 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

### Subclasses

### Superclasses

### References

^{1)}G. Bruns and J. Harding,

**, Order 14 (1997/98), no. 3, 193–209 MRreview**

*Amalgamation of ortholattices*Trace: » ortholattices