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The set $\{1,2,3,4,6,8,12,24\}$ together with the two binary operations LCM and GCD represents which of the following algebraic structure?

1. group
2. ring
3. lattice
4. field

### 1 comment

Only groups is there in gate syllabus . Other options aren't there as per syllabus ..
Will such questions come in exam ?

If you construct a Hasse Diagram for the given set with the relation x divides y,

you will find that LCM of two numbers will be the least upper bound of those two elements and GCD will be the greatest lower bound of the elements and for every two elements we have Infimum and Supremum.

This is nothing but definition of a Lattice.

Hence option C.

plz explain..
You are correct. It is a lattice. I'll update the answer key and marks.

But can a lattice ever be a field?
I don't think so because as far as i understood, Ring is a set in which only + and . operations are defined,

and field has more axioms on these two operations only...but lattice is more abstract as it can be constructed for many operations(eg. set operations).

So lattice can be a field if it is defined on + and . operation i guess so.

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