How to encode Hasse diagram with 0's and 1's

401 Views Asked by At

Boolean algebra is defined as lattice which is isomorphic with $B_n$. I know how to construct Hasse diagram. But couldnt understand how to assign 0's and 1's to nodes. enter image description here.

Please tell the general mechanism for assigning boolean values to hasse diagram.

$B_n$ is defined as : If the Hasse diagram of the lattice corresponding to a set with $n$ elements is labeled by sequences of 0 and 1 of length $n$, then the resulting lattice is named $B_n$

1

There are 1 best solutions below

0
On

In your diagram nodes can be seen at various levels. Level $k$ means exactly $k$ 1's are present in the length $n$ string.

A slanted line joining two nodes means a single zero in the node below has been changed to 1.

For example in the last diagram by starting with $000000$ for the bottom most element one scans the row above left to right and replace that 0 into 1. So level 1 it will be $10000, 01000,00100,00010,00001$. Continue in the same way until you reach the top which is a node with all 1's. Thats all.