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## Bounded lattices

Abbreviation: **BLat**

### Definition

A ** 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

$\begin{array}{lr} f(1)= &1\\ f(2)= &1\\ f(3)= &1\\ f(4)= &2\\ f(5)= &5\\ \end{array}$ $\begin{array}{lr} f(6)= &15\\ f(7)= &53\\ f(8)= &222\\ f(9)= &1078\\ f(10)= &5994\\ \end{array}$ $\begin{array}{lr} f(11)= &37622\\ f(12)= &262776\\ f(13)= &2018305\\ f(14)= &16873364\\ f(15)= &152233518\\ \end{array}$ $\begin{array}{lr} f(16)= &1471613387\\ f(17)= &15150569446\\ f(18)= &165269824761\\ f(19)= &\\ f(20)= &\\ \end{array}$

\hyperbaseurl{http://math.chapman.edu/structures/files/}

### Subclasses

### Superclasses

### References

Trace: » bounded_lattices