### Table of Contents

## Complemented lattices

Abbreviation: **CdLat**

### Definition

A \emph{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}$