Complemented Lattice

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 ∨ b = 1 and a ∧ b = 0.

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 there is an element b such that

a ∨ b = d and a ∧ 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.