Determine whether, (P(S),*),where * stands for intersection is a semi-group, a monoid or neither. If it is a monoid, specify its identity.
let A,B∈ P(S) where A&B are subset of S.
As intersection of two subsets of S is subset of S.
Therefore A*(B*C)=A ∩ B ∈ P(S). Thus closure law holds in P(S).
For A,B,C∈ P(S)
Thus associative law holds and P(S)
And hence ( P(S),* ) is a semi-group.
For A∈P(S) where A is a subset of S we have S∈P(S) such that
Thus S is an identity element in P(S) . And hence ( P(S),* ) is a monoid.