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