### Table of Contents

## Distributive residuated lattices

Abbreviation: **DRL**

### Definition

A \emph{distributive residuated lattice} is a residuated lattice $\mathbf{L}=\langle L, \vee, \wedge, \cdot, e, \backslash, /\rangle$ such that

$\vee, \wedge$ are distributive: $x\wedge(y\vee z) =(x\wedge y) \vee (x\wedge z)$

Remark:

##### Morphisms

Let $\mathbf{L}$ and $\mathbf{M}$ be distributive residuated lattices. A morphism from $\mathbf{L}$ to $\mathbf{M}$ is a function $h:L\rightarrow M$ that is a homomorphism:

$h(x\vee y)=h(x)\vee h(y)$, $h(x\wedge y)=h(x)\wedge h(y)$, $h(x\cdot y)=h(x)\cdot h(y)$, $h(x\backslash y)=h(x)\backslash h(y)$, $h(x/y)=h(x)/h(y)$, $h(e)=e$

### Examples

Example 1:

### Basic results

### Properties

Classtype | variety |
---|---|

Equational theory | |

Quasiequational theory | undecidable |

First-order theory | undecidable |

Locally finite | no |

Residual size | unbounded |

Congruence distributive | yes |

Congruence modular | yes |

Congruence n-permutable | yes, n=2 |

Congruence regular | no |

Congruence e-regular | yes |

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)= &3

f(4)= &20

f(5)= &115

f(6)= &899

f(7)= &7782

f(8)= &80468

\end{array}$