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