Distributive lattice

Question

Q)which of the following is not a distributive lattice?

a) [P(A);$\preceq$ ] where  A = { a,b,c,d }

b) [ {1,2,3,5,30} ; / ]

Comments

A lattice is distributive if it does not contain these two lattices

Answer 1

in  distributed lattice each element has atmost one complement

in this lattice complement of 2 is 3 and 5. and hence it is not distributed lattice

Comments

please explain 1st options??

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power set is always distributed lattice. there is no comparable element. simillarlly for 4 element

Answer 2

i think option b). A propery that can be used to eliminate non-distributive lattice is that complement of element in lattice must be unique but here element 2,3,5 have 2 complements each.

In option a) P(A) power set of a we will have 2^n elemets. in hasse diagram we'll have 16 nodes and 32 (n * 2^(n-1)) edges. therefore this is a boolean lattice. Hence distributed bcoz every boolean lattice is distributed and complemented.
correct me if i'm wrong.

Comments

If lattice has to be distributed and complemented, so we say that boolean algebra.If it has to be boolean algebra then the number of elements in lattice diagram should be $2^{n}$ and number of edges should be $n.2^{n-1} $, Where n = number of elements.

please check if I'm wrong?

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@Lakshman ,u r right...A lattice is boolean algebra if and only if it is both complemented and distributed ..statement and its converse both r true here...A boolean algebra always have 2ⁿ elements and  n.2^n-1 edges..

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 ankitgupta.1729  is right

PS : you guys are answering so fast that i missed this question :p

Answer 3

 [P(A);⪯] where  A = { a,b,c,d }

In case of {a,b,c,d} the Hasse diagram of the Power Set will have no element with a complement, except the endpoints of the longest antichain.

=> Each element has at most one complement => Distributive lattice.

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[ {1,2,3,5,30} ; / ]

Multiple complements of a single element would be found in the Hasse diagram for it. So, not distributive lattice.