GATE CSE 1994 | Question: 2.9

Question

The Hasse diagrams of all the lattices with up to four elements are ________ (write all the relevant Hasse diagrams)

Comments

what is meant by ' up to 4 elements ' here..should we consider lattices with 3 elements and 2 elements also??

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We can have a single element as a lattice. Check here.

I guess this is the answer.

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yes right. discrete mathematics - Is a single element a Lattice? - Mathematics Stack Exchange

 

Answer 1

1. Diamond structure
2. Straight line structure

Comments

For straight line structure one of the relationship could be $cardinality$.

$|\phi| \leq |a| \leq |b| \leq |a,b|$

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one element also.

Answer 2

We cant draw a lattice with 1 element 

Comments

why can't  draw a lattice with one element???

plz give strong reason???

@Bikram sir

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I think we can draw one for a POSET with single element.

Consider a POSET = $\left[\{(2,2)\}, \leq\right]$

Of course, it has LUB, GLB $= 2$. So it's a lattice.

It will be a single dot.

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 discrete mathematics - Is a single element a Lattice? - Mathematics Stack Exchange

 

Answer 3

1 element lattice isn't possible. (Why?)

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For 2 elements, only a chain is possible.

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For 3 elements, only a chain is possible.

This is not a valid 3 element lattice. Because 2 $\leq$ 3 and 1$\leq$ 3 are both drawn. That should have been an implicit transitive edge => a chain.

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For 4 elements, chain and diamond.

Comments

Didn’t get that. Please explain someone.

Answer 4

One diamond diagram as u can see above and one like below

Chain diagram;

Like 1-->2--->3--->4....in vertical manner.