What does the complement mean as it relates to Boolean Algebra?

Question

For a lattice to be a Boolean Algebra it must be a distributive lattice and contain complements. What does the word complement mean?

Answer 1

A complemented lattice is a bounded lattice (with least element 0 and greatest element 1), in which every element a has a complement, i.e. an element $b$ such that

$$a \lor b = 1\quad\text{and} \quad a \land b = 0.$$

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In general an element may have more than one complement. However, in a bounded distributive lattice every element will have at most one complement. A lattice in which every element has exactly one complement is called a uniquely complemented lattice.

[A lattice with the property that every interval is complemented is called a relatively complemented lattice. In other words, a relatively complemented lattice is characterized by the property that for every element a in an interval [c, d] there is an element b such that

$$a \lor b = d\text{ and} \;\;a\land b = c.$$

Such an element b is called a complement of a relative to the interval.]

A distributive lattice is complemented if and only if it is bounded and relatively complemented.

Boolean algebras are a special case of orthocomplemented lattices, which in turn are a special case of complemented lattices (with extra structure). These structures are most often used in quantum logic, where the closed subspaces of a separable Hilbert space represent quantum propositions and behave as an orthocomplemented lattice.

Orthocomplemented lattices, like Boolean algebras, satisfy de Morgan's laws:

$$(a \lor b)^\perp = a^\perp \land b^\perp$$

$$(a \land b)^\perp = a^\perp \lor b^\perp.$$