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+15 votes

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
in Set Theory & Algebra by Boss (17.5k points)
edited by | 2k views

if any subset contains above structures, then it is not a distributive lattice. 

we know given hasse diagram is lattice and it containing above structue so it is not distributive. B


 That's because in the first figure, x has two complements.

In the second figure, y has two complements.

To be a distributive lattice, there must be at most one complement. :)

3 Answers

+24 votes
Best answer
Option B.

A lattice has lub and glb but to be distributive it should also have a unique complement.
by Active (3.3k points)
selected by
Yes it is a lattice should have unambiguous unique suprenum and infinum for all horizontal elements in the hasse diagram.
How to calculate complement can any1 plz explain.

Thank you in advance
shubhamdarokar try checking whether LHS and RHS are same in  $b \wedge (c \vee d)=(b\wedge c)\vee(b\wedge d)$. If yes, it is a distributive lattice, else not.

$\bullet$ It's a lattice as it's every pair of elements has LUB and GLB.

$\bullet$ It is not a distributive lattice as-
$\Rightarrow \ b ∧ (c ∨ d) = b ∧ a = b$
$(b ∧ c) ∨ (b ∧ d) = e ∨ e = e$
$b \neq e $

$\Rightarrow$ All 3 elements- b,c and d have 2 complements.

$\Rightarrow$ It's a famous diamond structure $(M_3)$ lattice which is non-distributive. Moreover, any lattice is distributive if and only if it does not contain $M_3 \ or \ N_5$ as sublattice.

$\bullet$ A lattice is a Boolean algebra if and only if it is distributive and complemented.
It is not distributive hence not boolean algebra. 


Regarding lattice I've two doubts,

  1. taking (b,c) it has upperbound 'a' & lowerbound 'e' then what is the LUB & GLB, Is 'a' & 'e' itself LUB & GLB?????
  2. Suppose if we take (e,d) then what will be the LUB & GLB????

please correct me where I'm lacking???


@MRINMOY_HALDER 1.LUB is Least(Minimum) element in Upper Bound and GLB is Greatest(Minimum) element in Lower Bound.

2. For (e,d) LUB is d and GLB is e.

It is a well known non-distributive lattice-The "Kite" lattice.
+9 votes

Option B is appropriate for it.

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

But It not Distributed bcz there exist more than complement of element.So it ever be Boolean algebra.  

by Boss (10.2k points)

@Paras Nath given Hasse diagram is totally different from poset  i think given reation is not '<' (less than ) it is other relation defined according to hasse diagram 

yes, it may be any relation. because nothing is mentioned about it. But, this can be answered by looking at hasse diagram only.
0 votes

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)
by Loyal (6.5k points)

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