Relation of divisibility {0,1,…,20} - Hasse diagram

Question

I am trying to draw a Hasse diagram of divisibility but AFAIK it's not correct.

I connected 4 with 8 , 12 and 20.

6 with 18 and 12,

5 with 15 and 10,

3 with 9, 6, 15 H 2 with 6, 4, 10 and 14.

1 with prime numbers

Is this correct? Thanks. The rest should correct.

Answer 1

You're missing many connections (each element should be connected to one that's greater and minimal among the greater elements). You're also forgetting about $0$.

At the lowest level you have to place the minimum, that is, $1$.

At the next level, the primes: $2$, $3$, $5$, $7$, $11$, $13$, $17$ and $19$.

Next level, the products of two (not necessarily distinct) primes, that is, $4$, $6$, $9$, $10$, $14$, $15$.

Next level, the products of three primes: $8$, $12$, $18$, $20$.

Last level, the maximum, that is, $0$.

Connections:

• $1$ is connected to every term at the next level (the primes);
• $2$ is connected to $4$, $6$, $10$;
• $3$ is connected to $6$, $9$, $15$;
• $5$ is connected to $10$, $15$, $20$;
• $7$ is connected to $14$;
• $11$, $13$, $17$, $19$ are connected to $0$;
• $4$ is connected to $8$, $12$, $20$;
• $6$ is connected to $12$, $18$;
• $10$ is connected to $20$;
• $8$, $12$, $18$, $20$ are connected to $0$.