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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} ; / ]

3 Answers

5 votes
5 votes

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

2 votes
2 votes
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.
edited by
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 [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.


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

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

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