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+8 votes

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

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
asked in Set Theory & Algebra by Veteran (18k points) | 1k 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

2 Answers

+16 votes
Best answer
Option b ,a lattice has lub and glb but to be distributive it should have unique complement.
answered by Loyal (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.
+6 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.  

answered by Boss (8.4k 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.

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