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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
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Ans is lattice (D).

For LCM and GCD:

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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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“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

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