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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 . A conditionally complete lattice satisfies at
The result is a distributive lattice and is used in Birkhoff's representation theorem. However, it may have many more elements than are needed to form a completion of S. [5] Among all possible lattice completions, the Dedekind–MacNeille completion is the smallest complete lattice with S embedded in it. [6]
A continuous lattice is a complete lattice that is continuous as a poset. An algebraic lattice is a complete lattice that is algebraic as a poset. Both of these classes have interesting properties. For example, continuous lattices can be characterized as algebraic structures (with infinitary operations) satisfying certain identities.
For example, the following is an equivalent law that avoids the use of choice functions [citation needed]. For any set S of sets, we define the set S # to be the set of all subsets X of the complete lattice that have non-empty intersection with all members of S. We then can define complete distributivity via the statement
In the mathematical areas of order and lattice theory, the Knaster–Tarski theorem, named after Bronisław Knaster and Alfred Tarski, states the following: Let (L, ≤) be a complete lattice and let f : L → L be an order-preserving (monotonic) function w.r.t. ≤ . Then the set of fixed points of f in L forms a complete lattice under ≤ .
The perfect double cover Co 0 of Co 1 is the automorphism group of the Leech lattice, and is sometimes denoted by ·0. Subgroup of Co 0; fixes a norm 4 vector in the Leech lattice. Subgroup of Co 0; fixes a norm 6 vector in the Leech lattice. It has a doubly transitive permutation representation on 276 points.
A poset is a complete lattice if and only if it is a cpo and a join-semilattice. Indeed, for any subset X, the set of all finite suprema (joins) of X is directed and the supremum of this set (which exists by directed completeness) is equal to the supremum of X. Thus every set has a supremum and by the above observation we have a complete lattice.
X is complete if and only if every bounded set that is closed in the order topology is compact. A totally ordered set (with its order topology) which is a complete lattice is compact. Examples are the closed intervals of real numbers, e.g. the unit interval [0,1], and the affinely extended real number system (extended real