### Table of Contents

## Integral residuated lattices

Abbreviation: **IRL**

### Definition

An \emph{integral residuated lattice} is a residuated lattice $\mathbf{L}=\langle L, \vee, \wedge, \cdot, 1, \backslash, /\rangle$ that is

\emph{integral}: $x\le 1$

##### Morphisms

Let $\mathbf{A}$ and $\mathbf{B}$ be integal residuated lattices. A morphism from $\mathbf{A}$ to $\mathbf{B}$ is a function $h:A\rightarrow B$ that is a homomorphism: $h(x\vee y)=h(x)\vee h(y)$, $h(x\wedge y)=h(x)\wedge h(y)$, $h(x\cdot y)=h(x)\cdot h(y)$, $h(x\backslash y)=h(x)\backslash h(y)$, $h(x/y)=h(x)/h(y)$, $h(1)=1$

### Examples

Example 1: The negative cone of any l-group, e.g., $\mathbb Z^-$

### Basic results

### Properties

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

Equational theory | decidable |

Quasiequational theory | decidable |

First-order theory | undecidable |

Locally finite | no |

Residual size | unbounded |

Congruence distributive | yes |

Congruence modular | yes |

Congruence $n$-permutable | yes |

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)= &2\\ f(4)= &9\\ f(5)= &49\\

\end{array}$ $\begin{array}{lr}

f(6)= &364\\ f(7)= &\\ f(8)= &\\ f(9)= &\\ f(10)= &\\

\end{array}$