### Table of Contents

## Bounded lattices

Abbreviation: **BLat**

### Definition

A \emph{bounded lattice} is a structure $\mathbf{L}=\langle L,\vee,0,\wedge,1\rangle$ such that

$\langle L,\vee,\wedge\rangle $ is a lattice

$0$ is the least element: $0\leq x$

$1$ is the greatest element: $x\leq 1$

##### Morphisms

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

$h(x\vee y)=h(x)\vee h(y)$, $h(x\wedge y)=h(x)\wedge h(y)$, $h(0)=0$, $h(1)=1$

### Examples

Example 1:

### Basic results

### Properties

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

Equational theory | decidable |

Quasiequational theory | decidable |

First-order theory | undecidable |

Congruence distributive | yes |

Congruence modular | yes |

Congruence n-permutable | no |

Congruence regular | no |

Congruence uniform | no |

Congruence extension property | no |

Definable principal congruences | no |

Equationally def. pr. cong. | no |

Amalgamation property | yes |

Strong amalgamation property | yes |

Epimorphisms are surjective | yes |

Locally finite | no |

Residual size | unbounded |

### Finite members

$\begin{array}{lr}
f(1)= &1

f(2)= &1

f(3)= &1

f(4)= &2

f(5)= &5

\end{array}$
$\begin{array}{lr}
f(6)= &15

f(7)= &53

f(8)= &222

f(9)= &1078

f(10)= &5994

\end{array}$
$\begin{array}{lr}
f(11)= &37622

f(12)= &262776

f(13)= &2018305

f(14)= &16873364

f(15)= &152233518

\end{array}$
$\begin{array}{lr}
f(16)= &1471613387

f(17)= &15150569446

f(18)= &165269824761

f(19)= &

f(20)= &

\end{array}$