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A semigroup is a pair where is a non-empty set and is an associative binary operation on A monoid is a semigroup with identity element in the sence that for any
Let us show that the set of positive integer is a monoid for the operation defined by
If then and hence the operation is defined on the set
for any we conclude that operation is associative, and hence is a semigroup.
Taking into account that for each we conclude that is the identity of the semigroup and consequently is a monoid.