lattice(redirected from Lattice theory)
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an algebraic concept. A lattice is a nonempty set S with two binary operations, join and meet, denoted by ⋃ and ⋂, respectively. In other words, to each pair of elements a and b in 5 there is associated a unique element a ⋃ b in S, their join, and a unique element a ⋂ b in 5, their meet. The operations satisfy the following axioms:
|(1) associativity||(a ⋃ b) ⋃ c = a ⋃ (b ⋃ c)|
|(a ⋂ b) ⋂ c = a ⋂ (b ⋂ c)|
|(2) commutativity||a ⋃ b = b ⋃ a|
|a ⋂ b = b ⋂ a|
|(3) absorption||(a ⋃ b) ⋂ a = a|
|(a ⋂ b) ⋃ a = a|
Examples of lattices include the set of positive integers with a ⋂ b the greatest common divisor and a ⋃ b the least common multiple of a and b, the set of all subsets of an arbitrary set with set theoretic union and intersection as the lattice operations, and the set of real numbers with a ⋃ b = max (a, b) and a ⋂ b = min (a, b).
Certain lattices satisfying additional requirements have been studied in great detail. Examples of such lattices are distributive lattices, modular (or Dedekind) lattices, and complemented lattices. A very important type of lattice is a Boolean algebra, that is, a distributive lattice with zero and one in which each element has a complement. Boolean algebras play an important role in mathematical logic and probability theory. Other kinds of lattices are used in set theory, topology, and functional analysis.
There is a natural way of using the lattice operations to partially order the lattice and thus establish the equivalence of lattice theory and the theory of partially ordered sets.
The concept of the lattice first appeared in the mid-19th century and was first fully defined by J. W. R. Dedekind.
REFERENCESBirkhoff, G. Teoriia struktur. Moscow, 1952. (Translated from English.)
Skorniakov, L. A. Elementy teorii struktur. Moscow, 1970.
Sikorski, R. Bulevy algebry. Moscow, 1969. (Translated from English.)
Vladimirov, D. A. Bulevy algebry. Moscow, 1969.
This definition has been standard at least since the 1930s and probably since Dedekind worked on lattice theory in the 19th century; though he may not have used that name.
See also complete lattice, domain theory.