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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 (

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?

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up vote 5 down vote accepted

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.

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