Search results
Results From The WOW.Com Content Network
An example is the Knaster–Tarski theorem, which states that the set of fixed points of a monotone function on a complete lattice is again a complete lattice. This is easily seen to be a generalization of the above observation about the images of increasing and idempotent functions.
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]
An ordered vector space is called order complete, Dedekind complete, a complete vector lattice, or a complete Riesz space, if it is order complete as a subset of itself, [1] [2] in which case it is necessarily a vector lattice. An ordered vector space is said to be countably order complete if each countable subset that is bounded above has a ...
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.
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
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.
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
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 ≤ .