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

$x\backslash wedge\; y=0$    and    $\backslash quad\; x\backslash vee\; y=1.$

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:

$x\; \backslash wedge\; (y\; \backslash vee\; z)\; =\; (x\; \backslash wedge\; y)\; \backslash vee\; (x\; \backslash wedge\; z),$

which is also bounded, then each element x will have at most one complement.

Similarly, in an orthocomplemented lattice it can be shown that each element has exactly one complement - in fact, there is an idempotent order-reversing function from elements to their complements.

Thus in a Boolean algebra, which is both a complemented distributive lattice and an orthocomplemented lattice, complements exist and are unique.

Lattice theory

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