is D36 distributive ?

Question

In one text I read that , if n is square free it is DISTRIBUTIVE
in other text I read that if n is square free  it is BOOLEAN ALGEBRA .

Which is most correct ?

Here D36 is not square free then... what conclusion can I make ?

Comments

We talk about distributivity and boolean algebra only iff it's a lattice(POSET). So, Relation needs to be provided with Set under discussion.

So, the correct question would go like : let $[D_{36},/]$ (or) Set $D_{36}$ on relation divides. (although its know that we talk about division mostly)

Answer 1

When we draw the  hasse diagram, 6 doesnt have a complement. Now, for a lattice to be distributive, every element must have unique complement.

Hence, not distributive.

Comments

In A distributive lattice, each element can have at most one complement, it could even have zero also.

Answer 2

You can simply check whether d36 is distributive  or not by checking whether we can get it by product of distinct primes for example
D30=2.3.5=30 possible  d40=2.2.2.5 not possible as 2 came 3 times not distinct d36=2.3.2.3 not distributive as we did not get it through product of distinct primes

Answer 3

D₃₆ is not a boolean algebra and not a distributive lattice .

 
36 = 3*2*2*3 so 2 and 3 is repeating hence it is not distributive .

We did not get 36 as product of distinct primes , hence it is not distributive .

Comments

@Bikram sir, then according to this D18 also would not be distributive as 18 = 2*3*3 and it does not have distinct primes . but on the internet it is given that D18 is distributive lattice.

kindly please clarify over this.