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+15 votes

Consider the set of integers $\{1,2,3,4,6,8,12,24\}$ together with the two binary operations LCM (lowest common multiple) and GCD (greatest common divisor). Which of the following algebraic structures does this represent?

  1. group
  2. ring
  3. field
  4. lattice
in Set Theory & Algebra by Veteran (106k points) | 1.2k views

2 Answers

+25 votes
Best answer

Ans is lattice (D).

For LCM and GCD:

by Veteran (63k points)
edited by
How did you draw this lattice? How two elements are related?
I think this lattice has been drawn like this:

For any two elements a and b in this lattice, if a < b then b is the LCM of a and b and a is the GCD of a and b. For example, if 1 and 3 are connected then 3 is the LCM of 1 and 3 and 1 is the GCD of 1 and 3.

This is lattice (D24,|)  over partial order relation divisibility. Where D24 indicates positive integral divisors of 24.

LCM is given by LUB and HCF is given by GLB.

how can we have two operations in a lattice?

The 2 operations are infimum and supremum
+12 votes

Ring is set with 2 operations such that $(A,+,*)$

 1. $(A,+)$ should be an abelian group.

 2. $(A,*)$ should be a semigroup.

 3. $*$ should be distributed over $+$

Field  $(A,+,*)$ :

 1. $(A,+)$ should be an abelian group.

 2. $(A-{e},*)$ should be an abelian group here $e$ is identity element.

 3. $*$ should be distributed over $+$.

 Group $(A,LCM):$

   1. Associative

   2. Should have an identity.

   3. Inverse.

In Question $A=\{1,2,3,4,6,8,12,14\}$ and operations are LCM , GCD

(A,LCM) is associative =  (a LCM b)LCM c = a LCM (b LCM c)

identity = 1.

but there does not exist Inverse of elements.

so this is not a group. that means it cant be abelian group then it also not be group and field so

Ans: Lattice(D)

by Active (3.7k points)
edited by

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