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A lattice is an abstract structure studied in the mathematical subdisciplines of order theory and abstract algebra.It consists of a partially ordered set in which every pair of elements has a unique supremum (also called a least upper bound or join) and a unique infimum (also called a greatest lower bound or meet).
A partially ordered set (poset for short) is an ordered pair = (,) consisting of a set (called the ground set of ) and a partial order on . When the meaning is clear from context and there is no ambiguity about the partial order, the set X {\displaystyle X} itself is sometimes called a poset.
The complete subgroup lattice for D4, the dihedral group of the square. This is an example of a complete lattice. In mathematics, a complete lattice is a partially ordered set in which all subsets have both a supremum and an infimum ().
Lattices, partial orders in which each pair of elements has a greatest lower bound and a least upper bound. Many different types of lattice have been studied; see map of lattices for a list. Partially ordered sets (or posets), orderings in which some pairs are comparable and others might not be
A partially ordered set that is both a join-semilattice and a meet-semilattice is a lattice. A lattice in which every subset, not just every pair, possesses a meet and a join is a complete lattice. It is also possible to define a partial lattice, in which not all pairs have a meet or join but the operations (when defined) satisfy certain axioms ...
For the partially ordered system of all subsets of a finite set, ordered by set inclusion, the antichains are called Sperner families and their lattice is a free distributive lattice, with a Dedekind number of elements. More generally, counting the number of antichains of a finite partially ordered set is #P-complete.
Consider a partially ordered set (X, ≤). As a first simple example, let 1 = {*} be a specified one-element set with the only possible partial ordering. There is an obvious mapping j: X → 1 with j(x) = * for all x in X. X has a least element if and only if the function j has a lower adjoint j *: 1 → X.
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 set with two associative, commutative idempotent binary operations linked by corresponding absorption laws.