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2 votes
2 votes

Which of the above lattice is distributve?

a) both iii and iv)

b) only iv)

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2 Answers

3 votes
3 votes

In first lattice, element $\color{green}{d}$ and $ \color{green}{ f}$ have Least Upper Bound and Greatest Lower Bound as $ \color{green}{ a}$ and $ \color{green}{ e}$ respectively, i.e, the upper bound and lower bound of lattice. And also $ \color{green}{ d}$ and $ \color{green}{ c}$ have $ \color{green}{ a}$ and $ \color{green}{ e}$ as LUB and GLB respectively. So, two complements  exist for element $ \color{green}{ d}$ in above lattice i.e, $ \color{green}{ f}$ and $ \color{green}{ c}$, Thus it cannot be a distributive lattice.

In second lattice no. of vertices $= 2^3 = 8$ and number of edges $= 3*2^{3-1} = 12$ and it is isomorphic to a Boolean Algebra of $Order-3$, Hence it is Both a Complemented and distributive lattice.

So, (b) is correct answer here.

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0 votes
first one is not distributive

second one is just like power set of three element form  lattice

hence it is distributive

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