Complemented lattice

From Freepedia

In the mathematical discipline of order theory, and in particular, in lattice theory, a complemented lattice is a bounded lattice, (that is it has a least element 0 and a greatest element 1), in which each element x has a complement, defined as an element y such that

<math>x\wedge y=0</math>    and    <math> \quad x\vee y=1.</math>

Uniqueness

In general an element x may have more than one complement. However in a distributive lattice, that is a lattice in which, for all x, y and z, the distributive law holds:

<math> x \wedge (y \vee z) = (x \wedge y) \vee (x \wedge z),</math>

which is also bounded, then each element x will have at most one complement. Thus in a Boolean algebra, which is a complemented distributive lattice, complements are unique.