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+9 votes
996 views
The Hasse diagrams of all the lattices with up to four elements are _____ (write all the relevant Hasse diagrams)
in Set Theory & Algebra by Veteran (52.2k points) | 996 views
0

Anything other than this?

+3
one more is straight line ryt ?
+2
Chain diagram;

Like 1-->2--->3--->4....in vertical manner.
+2
what is meant by ' up to 4 elements ' here..should we consider lattices with 3 elements and 2 elements also??

4 Answers

+21 votes
Best answer
1. Diamond structure
2. Straight line structure
by Veteran (60.8k points)
selected by
0
For straight line structure one of the relationship could be $cardinality$.

$|\phi| \leq |a| \leq |b| \leq |a,b|$
+3 votes

We cant draw a lattice with 1 element 

by Loyal (8k points)
0
Why can't we draw a lattice with one element ?
0
i think definition itself says  every pair has lub and glb so atleast 2 element should be there for pair
0

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

plz give strong reason???

@Bikram sir

0
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.
+1 vote

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


For 2 elements, only a chain is possible.


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.


For 4 elements, chain and diamond.

by Loyal (6k points)
0 votes
One diamond diagram as u can see above and one like below

Chain diagram;

Like 1-->2--->3--->4....in vertical manner.
by Boss (10.2k points)
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