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Concept
Complemented lattice
Summary
In the mathematical discipline of order 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 satisfying a ∨ b = 1 and a ∧ b = 0.
Complements need not be unique.
A relatively complemented lattice is a lattice such that every interval [c, d], viewed as a bounded lattice in its own right, is a complemented lattice.
An orthocomplementation on a complemented lattice is an involution that is order-reversing and maps each element to a complement. An orthocomplemented lattice satisfying a weak form of the modular law is called an orthomodular lattice.
In bounded distributive lattices, complements are unique. Every complemented distributive lattice has a unique orthocomplementation and is in fact a Boolean algebra.
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 (viewed as a sublattice) 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 [c, d] 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. The lattice of subspaces of a vector space provide an example of a complemented lattice that is not, in general, distributive.
De Morgan algebra
An orthocomplementation on a bounded lattice is a function that maps each element a to an "orthocomplement" a⊥ in such a way that the following axioms are satisfied:
Complement law a⊥ ∨ a = 1 and a⊥ ∧ a = 0.
Official source
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