Complemented lattice
From Wikipedia, the free encyclopedia
In the mathematical discipline of order theory, and in particular, in lattice theory, 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.[1] 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.
An orthocomplemented lattice is a lattice equipped with an involutive order-reversing function from elements to complements. A distributive orthocomplemented lattice is called a Boolean algebra.
 In the pentagon lattice N5, the node on the right-hand side has two complements. |
 The diamond lattice M3 admits no orthocomplementation. |
 The lattice M4 admits 3 orthocomplementations. |
 The hexagon lattice admits a unique orthocomplementation, but it is not uniquely complemented. |
[edit] Notes
- ^ Rutherford (1965), Th.9.3 p.25.
[edit] References
- Rutherford, Daniel Edwin (1965). Introduction to Lattice Theory. Oliver and Boyd.
