TIFR CSE 2014 | Part B | Question: 16

Question

Consider the ordering relation $x\mid y \subseteq N \times N$ over natural numbers $N$ such that $x \mid y$ if there exists $z \in N$ such that $x ∙ z = y$. A set is called lattice if every finite subset has a least upper bound and greatest lower bound. It is called a complete lattice if every subset has a least upper bound and greatest lower bound. Then,

• A. $\mid$ is an equivalence relation.
• B. Every subset of $N$ has an upper bound under $|$.
• C. $\mid$ is a total order.
• D. $(N, \mid)$ is a complete lattice.
• E. $(N, \mid)$ is a lattice but not a complete lattice.

Comments

It's nothing but ( D_n,| ) // Lattice under divisibility relation.

Partial order relation : Yes

Need not be complete lattice.

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@Deepakk Poonia (Dee) 

Lattice as per que is for " Every subset" , but it has different definition in K.H Rosen.

Are they different and if yes , then which one to be considered ? 

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@Mk Utkarsh 

will u help me in this ?

1. Every subset can have a common element which they divide. ( lcm) 

2. Every element have element that devides i.e "1"       ( gcd)

then why not complete order ? And i too dont understood the argument about inclusion and exclusion of "0" make difference, how ?

 

( or is it due to unboundedness ?)

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Why do we  say that just because 0 is included
its a complete lattice . I agree every infinite subset will have a
a upper bound if 0 is included.But that doesnt mean that they will have lowest upper bound. Why isnt this possibility being proved when we say something is a complete lattice?

Answer 1

IT takes million of words to write a story but one picture to say it all

A complete lattice has every subset as a lattice

example

Set of all transitive relations on a set {1,2}