Completely distributive lattice

In the mathematical area of order theory, a completely distributive lattice is a complete lattice in which arbitrary joins distribute over arbitrary meets. Formally, a complete lattice L is said to be completely distributive if, for any doubly indexed family {xj,k | j in J, k in Kj} of L, we have where F is the set of choice functions f choosing for each index j of J some index f(j) in Kj. Complete distributivity is a self-dual property, i.e. dualizing the above statement yields the same class of complete lattices. Without the axiom of choice, no complete lattice with more than one element can ever satisfy the above property, as one can just let xj,k equal the top element of L for all indices j and k with all of the sets Kj being nonempty but having no choice function. Various different characterizations exist. For example, the following is an equivalent law that avoids the use of choice functions. 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 The operator ( )# might be called the crosscut operator. This version of complete distributivity only implies the original notion when admitting the Axiom of Choice. In addition, it is known that the following statements are equivalent for any complete lattice L: L is completely distributive. L can be embedded into a direct product of chains [0,1] by an order embedding that preserves arbitrary meets and joins. Both L and its dual order Lop are continuous posets. Direct products of [0,1], i.e. sets of all functions from some set X to [0,1] ordered pointwise, are also called cubes. Every poset C can be completed in a completely distributive lattice. A completely distributive lattice L is called the free completely distributive lattice over a poset C if and only if there is an order embedding such that for every completely distributive lattice M and monotonic function , there is a unique complete homomorphism satisfying .

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Distributivity (order theory)

In the mathematical area of order theory, there are various notions of the common concept of distributivity, applied to the formation of suprema and infima. Most of these apply to partially ordered sets that are at least lattices, but the concept can in fact reasonably be generalized to semilattices as well. Probably the most common type of distributivity is the one defined for lattices, where the formation of binary suprema and infima provide the total operations of join () and meet ().

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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). An example is given by the power set of a set, partially ordered by inclusion, for which the supremum is the union and the infimum is the intersection.

Distributive lattice

In mathematics, a distributive lattice is a lattice in which the operations of join and meet distribute over each other. The prototypical examples of such structures are collections of sets for which the lattice operations can be given by set union and intersection. Indeed, these lattices of sets describe the scenery completely: every distributive lattice is—up to isomorphism—given as such a lattice of sets. As in the case of arbitrary lattices, one can choose to consider a distributive lattice L either as a structure of order theory or of universal algebra.

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