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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 ?
asked in Set Theory & Algebra by Veteran (21.3k points)   | 308 views

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)

5 Answers

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Best answer
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
answered by Ajay singh  
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Dn if n is a square free number,then  it will be a boolean algebra because if it is perfect square ,then its factors will repeat and then it may lead to more than one complement of an element which is actually not a boolean algebra. ...so D36 is not a boolean algebra.
answered by Veteran (10.5k points)  
How do you account for distributive ?
+1 vote
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.
answered by Veteran (15.4k points)  
+1 vote

In Dn ,if n is square free number then it will be a boolean algebra along with the numbe of vertex should be 2^n and number of edges should be 2*2^n-2.

Above is most important condition to identify whether a relation is boolean algebra or not.Above rule is not for distributive lattice.

Distributive lattice fallow the distributive properties and sublattice properties.

Example:: D64 not Boolean algebra but D110 is boolean algebra.

answered by Boss (5.4k points)  
0 votes

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 .

answered by Veteran (44.7k points)  


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