Complemented lattice
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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>
[edit] 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.[1]
Similarly, in an orthocomplemented lattice it can be shown that each element has exactly one complement - in fact, there is an involutive 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.
[edit] Notes
- ^ Rutherford (1965), Th.9.3 p.25.
[edit] References
- Rutherford, Daniel Edwin (1965). Introduction to Lattice Theory. Oliver and Boyd.
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