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The following is the Hasse diagram of the poset $\left[\{a,b,c,d,e\},≺\right]$

The poset is :

  1. not a lattice
  2. a lattice but not a distributive lattice
  3. a distributive lattice but not a Boolean algebra
  4. a Boolean algebra
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4 Answers

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30 votes

Option B.

A lattice has a least upper bound (lub) and a greatest upper bound (glb), but to be distributive every element of the lattice should have at most one complement.Here, elements $b,c,d$ are complements of each other and hence the given lattice is not distributive.

Ref: https://math.stackexchange.com/questions/2814774/example-of-a-lattice-which-has-at-most-1-complement-for-its-every-element-but-it

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Option B is appropriate for it.

It is lattice bcz both LUB and GLB exist for each pair of the vertex in the above Hasse diagram.

But It is not Distributed bcz there exist more than a complement of element. So it ever be Boolean algebra.
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A lattice is a distributive lattice if each element has at most one complement.

Complements of b are c and d.

So, not distributive.

 

It sure is a lattice, though. So, Option B


 

Why is it not a Boolean algebra?

  • If every element of the lattice has at least one complement => Complemented lattice.
     
  • If every element of the lattice has strictly either 0 or 1 complement => Distributive lattice.
     
  • A complemented distributed lattice is a necessary condition for being Boolean algebra. (So Option D is False)
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