# identity and inverse/complement elements in a boolean algebra

In a boolean algebra, $$0$$ (the lattice's bottom) is the identity element for the join operation $$\lor$$, and $$1$$ (the lattice's top) is the identity element for the meet operation $$\land$$. For an element in the boolean algebra, its inverse/complement element for $$\lor$$ is wrt $$1$$ and its inverse/complement element for $$\land$$ is wrt $$0.$$

A Boolean algebra can be defined to be a complemented lattice that is also distributive. For a distributive lattice, the complement of $$x,$$ when it exists, is unique. See Wikipedia (http://en.wikipedia.org/wiki/Lattice_(order)#Complements_and_pseudo-complements).

The power set of a set $$S$$ is an example of Boolean algebra. $$S$$ is the identity for union and $$\emptyset$$ is the identity for intersection. However, for union, the complement of a set wrt $$S$$ is not unique; For intersection, the complement of a set wrt $$\emptyset$$ is not unique either. So is this a contradiction?

Thanks and regards!

By definition, $\rm b\:$ is a complement of $\rm a\:$ if $\rm\ a\vee b = 1,\ a\wedge b = 0\:$. So a unique complement must be a unique solution to both$\$ equations (involving both$\$ operations), not just a single operation - as you consider above. So there is no contradiction.