Is the number of sub-boolean algebras of a set with size of n equal to Bell(n)?

In boolean algebra (P(S),+,.,’) we must have S as 1 and {} as 0 in every possible sub-boolean algebra to hold id elements. We must have S-x for every subset x⊆S to hold complements. It seems like counting every possible partitions in S which is Bell(|S|) if i was not wrong. For example the number of possible sub-boolean algebra of ⟨p({a,b,c,d}),∪,∩⟩ is Bell(4)=15 , is it right ? If it is how we can define a bijection between every n class partitions to boolean algebra with size n ?