### Table of Contents

## Commutative residuated lattices

Abbreviation: **CRL**

### Definition

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

$\cdot$ is commutative: $xy=yx$

Remark:

##### Morphisms

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

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

### Examples

Example 1:

### Basic results

### Properties

Classtype | Variety |
---|---|

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, n=2 |

Congruence regular | No |

Congruence e-regular | Yes |

Congruence uniform | No |

Congruence extension property | Yes |

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)= &16

f(5)= &100

f(6)= &794

f(7)= &7493

f(8)= &84961

\end{array}$

### Subclasses

### Superclasses

Commutative multiplicative lattices

Commutative residuated join-semilattices