## Bounded distributive lattices

Abbreviation: **BDLat**

### Definition

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

$\langle L,\vee ,\wedge \rangle $ is a distributive 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 distributive lattices. A morphism from $\mathbf{L}$ to $\mathbf{M}$ is a function $h:L\to M$ 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: $\langle \mathcal P(S), \cup, \emptyset, \cap, S\rangle$, the collection of subsets of a set $S$, with union, empty set, intersection, and the whole set $S$.

### 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 | yes |

Definable principal congruences | no |

Equationally def. pr. cong. | no |

Amalgamation property | yes |

Strong amalgamation property | no |

Epimorphisms are surjective | no |

Locally finite | yes |

Residual size | 2 |

### Finite members

$\begin{array}{lr} f(1)= &1\\ f(2)= &1\\ f(3)= &1\\ f(4)= &2\\ f(5)= &3\\ \end{array}$ $\begin{array}{lr} f(6)= &5\\ f(7)= &8\\ f(8)= &15\\ f(9)= &26\\ f(10)= &47\\ \end{array}$ $\begin{array}{lr} f(11)= &82\\ f(12)= &151\\ f(13)= &269\\ f(14)= &494\\ f(15)= &891\\ \end{array}$ $\begin{array}{lr} f(16)= &1639\\ f(17)= &2978\\ f(18)= &5483\\ f(19)= &10006\\ f(20)= &18428\\ \end{array}$

Values known up to size 49 ^{1)}.

### Subclasses

### Superclasses

### References

^{1)}Marcel Erne, Jobst Heitzig and J\ā€¯urgen Reinhold,

**, Electron. J. Combin.,**

*On the number of distributive lattices***9**, 2002, Research Paper 24, 23 pp. (electronic)

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