### Table of Contents

## Bounded residuated lattices

Abbreviation: **RLat$_b$**

### Definition

A \emph{bounded residuated lattice} is a residuated lattice that is bounded:

$\bot$ is the least element: $\bot\vee x=x$

$\top$ is the greatest element: $\top\vee x=\top$

##### Morphisms

Let $\mathbf{A}$ and $\mathbf{B}$ be bounded residuated lattices. A morphism from $\mathbf{A}$ to $\mathbf{B}$ is a residuated lattice homomorphism $h:A\rightarrow B$ that preserves the bounds: $h(\bot)=\bot$ and $h(\top)=\top$.

### 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 | 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)= &\\ f(3)= &\\ f(4)= &\\ f(5)= &\\

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

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

\end{array}$

### Subclasses

[[...]] subvariety

[[...]] expansion

### Superclasses

[[...]] supervariety

[[...]] subreduct