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

Finite members

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

Subclasses

Superclasses

References


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