in Set Theory & Algebra edited by
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26 votes
26 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 edited by
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3 Answers

31 votes
31 votes
Best answer

Ans is lattice (D).

For LCM and GCD:

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4 Comments

There is no inverse for the set.

Identity of LCM : 1

Identity of GCD : 24

For Inverse : LCM(8,  x) = 1 , but there is no such x in the set

similarly,      : GCD(8 , x)=24 ,but there is no such x 

So, it is not a group. Therefore it is not both field and ring.

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To know why answer is “Lattice”, Refer below :

https://gateoverflow.in/43580/gate-cse-1992-question-14b?show=362531#a362531

To know why answer is not ring or field or group, Refer below :

https://gateoverflow.in/43580/gate-cse-1992-question-14b?show=261223#a261223

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@Deepak Sir, is this in syllabus now?
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25 votes
25 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)

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1 comment

If we consider 24 to be the identity element.
For HCF, 1 is inverse and 24 is identity.
For LCM, 24 is inverse and 1 is identity.
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10 votes
10 votes

“Lattice” can be defined in two “equivalent” ways : 

1. From POSET(Relation) point of view

2. From Algebra point of view.

Both definitions are equivalent in the sense that one definition implies another.

We already know Lattices as Posets. Let us see Lattices as Algebraic Structures.

Lattice as algebraic structures :

A lattice can also be defined as an algebra $(L,∧,∨)$ on a set L with two binary operations ∧ (meet) and ∨ (join). The algebra satisfies the following identities :

  1. Commutativity: $a∧b=b∧a; a∨b=b∨a$.
  2. Associativity: $(a∧b)∧c=a∧(b∧c); (a∨b)∨c=a∨(b∨c)$.
  3. Idempotence: $a∧a=a; a∨a=a$.
  4. Absorption: $a∧(a∨b)=a; a∨(a∧b)=a$,

where $a,b,c∈L$.


Basically, An algebraic structure $(L,∧,∨)$, consisting of a set $L$ and two binary operations $(∧,∨)$ , on $L$ is a lattice if “Commutative, Associative, Absorption, Idempotence” Properties are satisfied.

(Also NOTE that “Closure” property is implicit in the definition of “binary operation”, So, it need not be mentioned explicitly that it must be satisfied ; Also the idempotence property is redundant as it is implied by other properties)  

Now, Coming to the given question, The given set with two binary operations LCM, GCD ; satisfies ALL the mentioned properties, So, it is a Lattice.

https://www.math24.net/lattices

https://en.wikipedia.org/wiki/Lattice_(order)

https://mathworld.wolfram.com/Lattice.html

https://ncatlab.org/nlab/show/lattice

https://math.stackexchange.com/questions/1158018/definition-of-algebraic-structure

1 comment

Thank You Sir for this information..
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