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The binary operation defined on a,b∈z such that a*b= min⁡(a,b)then (A,*) is

a)monoid

b)group

c)algebric-structure

d)semi group

1 Answer

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For Checking if $(\mathbb{Z},*)$ is a semi-group we need to check 2 conditions

1) $\mathbb{Z}$ is closed under * 
     True
     "
A set is closed under some operation if applying the operation on any elements of the set gives an element which is still in that set"

2) $*$ is an associative operation 
    True
 Let us assume a<b<c | a,b,c $\in \mathbb{Z}$

$(a*b)*c = a*(b*c)$
$a*c = a*b$
$a = a$

Hence it is Semi-group

A monoid is a semi-group with an identity.

There exist no identity for operation (min) in set of integers because there is no minimum element in $\mathbb{Z}$ to satisfy.
Hence it is not a monoid 

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