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

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.

• What is your diagram supposed to show? All the numbers you've listed does divide each other as described, but it's not all division relationships in that set, but "The rest should correct." could cover the missing divisors, meaning that what you've done is correct. – Henrik Nov 18 '18 at 15:53

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 $$0$$.