Scott domains

In the mathematical field of order theory, a partially ordered set is bounded complete if all of its subsets which have some upper bound also have a least upper bound. Such a partial order can also be called consistently complete, since any upper bound of a set can be interpreted as some consistent (non-contradictory) piece of information that extends all the information present in the set. Hence the presence of some upper bound in a way guarantees the consistence of a set. Bounded completeness then yields the existence of a le ...

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As a natural consequence of the fact that semilattices are among the most basic "lattice-like" structures, they can be characterized both in terms of order theory and of universal algebra. Each of these descriptions is given below. Semilattice - Semilattices as posets. Consider a partially ordered set (S, ≤). S is a meet-semilattice if for all elements x and y of S, the greatest lower bound (mee ...

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Semilattice, Semilattice - Formal definition, Semilattice - Semilattices as posets, Semilattice - Semilattices as algebraic structures, Semilattice - Connection between both definitions, Semilattice - Examples, Semilattice - Morphisms of semilattices, Semilattice - Distributive semilattices, Semilattice - Complete semilattices, Semilattice - Free semilattices, Semilattice - Literature

Read more here: » Semilattice: Encyclopedia II - Semilattice - Formal definition

In this section, the central concepts and definitions of domain theory will be introduced. The above intuition of domains being information orderings will be emphasized to motivate the mathematical formalization of the theory. The precise formal definitions are to be found in the dedicated articles for each concept. A list of general order-theoretic definitions which include domain theoretic notions as well can be found in the order theory glossary. The most important concepts of domain theory will nonetheless be introduced below.

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Domain theory, Domain theory - Motivation and intuition, Domain theory - A guide to the formal definitions, Domain theory - Directed sets as converging specifications, Domain theory - Computations and domains, Domain theory - Approximation and finiteness, Domain theory - Bases of domains, Domain theory - Special types of domains, Domain theory - Important results, Domain theory - Literature

Read more here: » Domain theory: Encyclopedia II - Domain theory - A guide to the formal definitions

Today, the term "complete semilattice" is not a generally established notion and various inconsistent definitions exist. The main reason for this is that the obvious requirement of the existence of all (finite and infinite) joins and meets, respectively, immediately leads to partial orders that are in fact complete lattices. For an explanation why the presence of all joins entails the existence of all meets (and vice ve ...

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Semilattice, Semilattice - Formal definition, Semilattice - Semilattices as posets, Semilattice - Semilattices as algebraic structures, Semilattice - Connection between both definitions, Semilattice - Examples, Semilattice - Morphisms of semilattices, Semilattice - Distributive semilattices, Semilattice - Complete semilattices, Semilattice - Free semilattices, Semilattice - Literature

Read more here: » Semilattice: Encyclopedia II - Semilattice - Complete semilattices

In various situations, free semilattices exist. For example, the forgetful functor from the category of join-semilattices (and their homomorphisms) to the category of sets (and functions) admits a left adjoint. Therefore, the free join-semilattice F(S) over a set S is constructed by taking the collection of all non-empty finite subsets of S, ordered by subset inclusion. Clearly, S can be embedded into F(S) by a mapping e that takes any element s in S to the singleto ...

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Semilattice, Semilattice - Formal definition, Semilattice - Semilattices as posets, Semilattice - Semilattices as algebraic structures, Semilattice - Connection between both definitions, Semilattice - Examples, Semilattice - Morphisms of semilattices, Semilattice - Distributive semilattices, Semilattice - Complete semilattices, Semilattice - Free semilattices, Semilattice - Literature

Read more here: » Semilattice: Encyclopedia II - Semilattice - Free semilattices

It may be somewhat surprising that there is indeed an established notion of "distributivity" for semilattices, since one usually considers distributivity as an interaction of two operations. Yet, it turns out that there is a convenient generalization of the distributivity condition for lattices, which can be stated in presence of a single operation. See the article on distributivity (order theory) for a discussion ...

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Semilattice, Semilattice - Formal definition, Semilattice - Semilattices as posets, Semilattice - Semilattices as algebraic structures, Semilattice - Connection between both definitions, Semilattice - Examples, Semilattice - Morphisms of semilattices, Semilattice - Distributive semilattices, Semilattice - Complete semilattices, Semilattice - Free semilattices, Semilattice - Literature

Read more here: » Semilattice: Encyclopedia II - Semilattice - Distributive semilattices

Considering the algebraic definition above, it is easily seen what morphisms between semilattices should be considered: given two join-semilattices (S,) and (T,), a homomorphisms of (join-) semilattices is a function f : S → T with the property that f(xy) = f(x) f(y), i.e. f is just a homomorphism of the two semigroups. If the join-semilattices are furthermore equipped with a least element 0, then f should also be a morphism of monoids, i.e. one addit ...

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Semilattice, Semilattice - Formal definition, Semilattice - Semilattices as posets, Semilattice - Semilattices as algebraic structures, Semilattice - Connection between both definitions, Semilattice - Examples, Semilattice - Morphisms of semilattices, Semilattice - Distributive semilattices, Semilattice - Complete semilattices, Semilattice - Free semilattices, Semilattice - Literature

Read more here: » Semilattice: Encyclopedia II - Semilattice - Morphisms of semilattices

The primary motivation for the study of domains, which was initiated by Dana Scott in the late 1960s, was the search for a denotational semantics of the lambda calculus. In this formalism, one considers "functions" specified by certain terms in the language. In a purely syntactic way, one can go from simple functions to functions that take other functions as their input arguments. Using again just the syntactic transformations available in this formalism, one can obtain so called fixed point combinators (also called Y combinators); these, by definition, have the property that f(Y(< ...

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Domain theory, Domain theory - Motivation and intuition, Domain theory - A guide to the formal definitions, Domain theory - Directed sets as converging specifications, Domain theory - Computations and domains, Domain theory - Approximation and finiteness, Domain theory - Bases of domains, Domain theory - Special types of domains, Domain theory - Important results, Domain theory - Literature

Read more here: » Domain theory: Encyclopedia II - Domain theory - Motivation and intuition

A poset D is a dcpo iff each chain in D has a supremum. However, directed sets are strictly more powerful than chains. A poset D with a least element is a dcpo iff every monotone function f on D has a fixed point. If f is continuous then it has even a least fixed point, given as the least upper bound of all finite iterations of f on the least element 0: Vn in N f n(0). Of course, there are many other important results, depending on the application area where domain theory is to be appli ...

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Domain theory, Domain theory - Motivation and intuition, Domain theory - A guide to the formal definitions, Domain theory - Directed sets as converging specifications, Domain theory - Computations and domains, Domain theory - Approximation and finiteness, Domain theory - Bases of domains, Domain theory - Special types of domains, Domain theory - Important results, Domain theory - Literature

Read more here: » Domain theory: Encyclopedia II - Domain theory - Important results