In the mathematical discipline of order theory, and in particular, in lattice theory, a complemented lattice is a bounded lattice in which each element x has a complement, defined as a unique element ~x such that History Main article: History of mathematics In addition to recognizing how to count concrete objects, prehistoric peoples also recognized how to count abstract quantities, like time -- days, seasons, years. ... Order theory is a branch of mathematics that studies various kinds of binary relations that capture the intuitive notion of a mathematical ordering. ... The term lattice derives from the shape of the Hasse diagrams that result from depicting these orders. ... The term lattice derives from the shape of the Hasse diagrams that result from depicting these orders. ...
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A Boolean algebra may be defined as a complemented distributive lattice. In mathematics and computer science, Boolean algebras, or Boolean lattices, are algebraic structures which capture the essence of the logical operations AND, OR and NOT as well as the corresponding set theoretic operations intersection, union and complement. ... In mathematics, distributive lattices are lattices for which the operations of join and meet distribute over each other. ...
Every lattice can be generated from a basis for the underlying vector space by considering all linear combinations with integral coefficients.
In another mathematical usage, a lattice is a partially ordered set in which all nonempty finite subsets have a least upper bound and a greatest lower bound (also called supremum and infimum, respectively).
The lattice of submodules of a module and the lattice of normal subgroups of a group have the special property that x v (y ^ (x v z)) = (x v y) ^ (x v z) for all x, y and z in the lattice.
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