### Table of Contents

## Complemented modular lattices

Abbreviation: **CdMLat**

### Definition

A \emph{complemented modular lattice} is a complemented lattices $\mathbf{L}=\langle L,\vee ,0,\wedge ,1\rangle $ that is

modular lattices: $(( x\wedge z) \vee y) \wedge z=( x\wedge z) \vee ( y\wedge z) $

##### Morphisms

Let $\mathbf{L}$ and $\mathbf{M}$ be complemented modular lattices. A morphism from $\mathbf{L}$ to $\mathbf{M}$ is a function $h:L\to M$ 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:

### Basic results

This class generates the same variety as the class of its finite members plus the non-desargean planes.

### Properties

Classtype | first-order |
---|---|

Equational theory | decidable |

Quasiequational theory | undecidable |

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 | |

Definable principal congruences | |

Equationally def. pr. cong. | |

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)= &1

f(6)= &

f(7)= &

f(8)= &

\end{array}$