Hasse diagram of a complemented lattice. A point p and a line l of the Fano plane are complements if and only if p does not lie on l.
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.
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theory, a complementedlattice is a bounded lattice (with least element 0 and greatest element 1), in which every element a has a complement, i.e. an element...
diameters of these hyperbolas are hyperbolic-orthogonal. ComplementedlatticeComplemented subspace Hilbert projection theorem – On closed convex subsets...
element is the trivial topology. The lattice of topologies on a set X {\displaystyle X} is a complementedlattice; that is, given a topology τ {\displaystyle...
orthocomplemented lattice is complemented. (def) 8. A complementedlattice is bounded. (def) 9. An algebraic lattice is complete. (def) 10. A complete lattice is bounded...
Bounded lattice: a lattice with a greatest element and least element. Complementedlattice: a bounded lattice with a unary operation, complementation, denoted...
Boolean algebra: a complemented distributive lattice. Either of meet or join can be defined in terms of the other and complementation. Module: an abelian...
cyclic. Groups whose lattice of subgroups is a complementedlattice are called complemented groups (Zacher 1953), and groups whose lattice of subgroups are...
particular, any bounded lattice can be endowed with both a meet- and a join- monoid structure. The identity elements are the lattice's top and its bottom,...
ring Ring theory Lattice-like Lattice Semilattice Complementedlattice Total order Heyting algebra Boolean algebra Map of latticesLattice theory Module-like...
ring Ring theory Lattice-like Lattice Semilattice Complementedlattice Total order Heyting algebra Boolean algebra Map of latticesLattice theory Module-like...
realm of group theory, the term complemented group is used in two distinct, but similar ways. In (Hall 1937), a complemented group is one in which every subgroup...
theory, a pseudocomplement is one generalization of the notion of complement. In a lattice L with bottom element 0, an element x ∈ L is said to have a pseudocomplement...
ring Ring theory Lattice-like Lattice Semilattice Complementedlattice Total order Heyting algebra Boolean algebra Map of latticesLattice theory Module-like...
ring Ring theory Lattice-like Lattice Semilattice Complementedlattice Total order Heyting algebra Boolean algebra Map of latticesLattice theory Module-like...
ring Ring theory Lattice-like Lattice Semilattice Complementedlattice Total order Heyting algebra Boolean algebra Map of latticesLattice theory Module-like...
a Heyting algebra (also known as pseudo-Boolean algebra) is a bounded lattice (with join and meet operations written ∨ and ∧ and with least element 0...
matroid. Geometric lattices are complemented, and because of the interval property they are also relatively complemented. Every finite lattice is a sublattice...
ring Ring theory Lattice-like Lattice Semilattice Complementedlattice Total order Heyting algebra Boolean algebra Map of latticesLattice theory Module-like...
ring Ring theory Lattice-like Lattice Semilattice Complementedlattice Total order Heyting algebra Boolean algebra Map of latticesLattice theory Module-like...
ring Ring theory Lattice-like Lattice Semilattice Complementedlattice Total order Heyting algebra Boolean algebra Map of latticesLattice theory Module-like...
A lattice is a partially ordered set that is both a meet- and join-semilattice with respect to the same partial order. Algebraically, a lattice is a...
AK is an R-subalgebra that is a lattice. In general, there are a lot fewer orders than lattices; e.g., 1/2Z is a lattice in Q but not an order (since it...
inverse. At the same time, it is a generalization of bounded distributive lattices. The smallest semiring that is not a ring is the two-element Boolean algebra...
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