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Let $L$ be a set with a relation $R$ which is transitive, anti-symmetric and reflexive and for any two elements $a, b \in  L$, let the least upper bound $lub (a, b)$ and the greatest lower bound $glb (a, b)$ exist. Which of the following is/are true?

  1. $L$ is a poset

  2. $L$ is a Boolean algebra

  3. $L$ is a lattice

  4. None of the above

asked in Set Theory & Algebra by Veteran (59.7k points)
edited by | 860 views

3 Answers

+8 votes
Best answer

Which of the following is/are true? This is question with Multiple answers.

As our Relation $R$ on Set $L$ is Reflexive , anti symmetric & Transitive it is poset.

Since $LUB$ & $GLB$ exists for any two elements it is lattice.

Answer -> $A$ & $C$.

$B$ is not guaranteed to be true.

Ref: http://uosis.mif.vu.lt/~valdas/PhD/Kursinis2/Sasao99/Chapter2.pdf

answered by Boss (43.2k points)
edited by
+1
sir i think C only should be the answer as Lattice in itself denotes that it is a poset... So a more strict answer should be lattice. Correct me if i am wrong :D
+1
The Question says LUB and GLB exists for every pair.
It doesn't say "uniquely".....implying there can be a pair (a,b) for which more than one LUB/GLB can exist---->Not a Lattice.

Definitely a Poset.
Correct?
+1
A poset is lattice if it contain lub and gub for every pair, means if a graph is lattice then is also poset so that anwer will be C.
0
I think strongest answer is Lattice. Answer C
0
Hello Amitabh

Can there be more than one LCM or GCD for any pair (a,b)?

They are called least and greatest on the first place hence they must be unique.

So either a pair (a,b) won't have GLB/LUB or if they have then that must be unique.
0

Amitabh Tiwari 1

if LUB/GLB exists then it implies it's unique, if there exist more than one lower bound or upper bound, we say LUB/GLB doesn't exist.

+3 votes

it is given that set with a relation Relation R which is transitive, antisymmetric and reflexive so it is already a poset .Additionally, it is guaranteeing about the existence  of LUB and GLB too.so now it is  : Poset + Having lub and Glb both , That is Lattice . so option A and C both r true .

For being option B true it needs some extra information that is :

A lattice can be Boolean Algebra only  when It is distributed lattice , complimented lattice and bounded lattice along with it will have to satisfy the properties of Boolean algebra . and we dont have any such info about this lattice .

counter of B is :

  it is poset, lattice but not B.A

answered by Active (3.2k points)
+2 votes
As it is Reflexive , anti-symmetric , transitive , it is poset.

Since lub and gub exist for any pair of elements, the poset becomes a lattice. So, C is the answer.
answered by Active (4.2k points)
+4
this question has multiple answers ? right ?

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