complete lattice

As mentioned above, lattices can be characterized both as posets and as algebraic structures. Both approaches and their relationship are explained below. Lattice order - Lattices as posets. Consider a poset (L, ≤). L is a lattice if for all elements x and y of L, the set {x, y} has both a least upper bound (join, or supremum) and a greatest l ...

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Lattice order, Lattice order - Formal definition, Lattice order - Lattices as posets, Lattice order - Lattices as algebraic structures, Lattice order - Connection between the two definitions, Lattice order - Examples, Lattice order - Morphisms of lattices, Lattice order - Properties of lattices, Lattice order - Completeness, Lattice order - Distributivity, Lattice order - Modularity, Lattice order - Continuity and algebraicity, Lattice order - Complements and pseudo-complements, Lattice order - Free lattices, Lattice order - Important lattice-theoretic notions

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In mathematics, directed complete partial orders and complete partial orders are special classes of partially ordered sets. These orders, called dcpos and cpos for short, are characterized by particular completeness properties. Both dcpos and cpos are considered in domain theory and have major applications in theoretical computer science and denotational semantics. Complete partial order - Definition. A partially ordered set is a directed complete partial order (dcpoIncluding:

• Complete partial order - Definition
• Complete partial order - Properties and related articles
• Complete partial order - Examples

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In mathematics, a continuous function is a function in which arbitrarily small changes in the input produce arbitrarily small changes in the output. If small changes in the input can produce a broken jump in the changes of the output (or the value of the output is not defined), the function is said to be discontinuous (or to have a discontinuity). The context in this entry is real-valued functions on the real domain or on topological or metric spaces other than the complex numbers; for complex-valued functions see comple ...

Including:

• Continuous function - Real-valued continuous functions
• Continuous function - Epsilon-delta definition
• Continuous function - Heine definition of continuity
• Continuous function - Examples
• Continuous function - Facts about continuous functions
• Continuous function - Continuous functions between metric spaces
• Continuous function - Continuous functions between topological spaces
• Continuous function - Continuous functions between partially ordered sets

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In mathematics and related technical fields, a mathematical object is complete if nothing needs to be added to it. This is made precise in various ways, several of which have a related notion of completion. It should be noted that "complete" here is just a term that takes on specific meanings in specific situations, and not every situation in which a type of "completion" occurs is called a "completion". See, for example, algebraically closed field, compactification, or Gödel's incompleteness theorem. A metric spac

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In mathematics, the unit interval is the interval [0,1], that is the set of all real numbers x such that zero is less than or equal to x and x is less than or equal to one. The unit interval plays a fundamental role in homotopy theory, a major branch of topology. It is a metric space, compact, contractible, path connected and locally path connected. As a topological space, it is homeomorphic to the extended real number line. The unit interval is a one-dimensional analytical manifold with boundary {0,1}, carrying a ...

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The appropriate notion of a morphism between two lattices can easily be derived from the algebraic definition above: given two lattices (L, , ) and (M, , ), a homomorphism of lattices is a function f : L → M with the properties that f(xy) = f(x) f(y), and f(xy) = f(x) f(y). Thus f is a homomorphism of the two underlying semilattices. If the lattices are ...

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Lattice order, Lattice order - Formal definition, Lattice order - Lattices as posets, Lattice order - Lattices as algebraic structures, Lattice order - Connection between the two definitions, Lattice order - Examples, Lattice order - Morphisms of lattices, Lattice order - Properties of lattices, Lattice order - Completeness, Lattice order - Distributivity, Lattice order - Modularity, Lattice order - Continuity and algebraicity, Lattice order - Complements and pseudo-complements, Lattice order - Free lattices, Lattice order - Important lattice-theoretic notions

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The definitions above already introduced the simple condition of being a bounded lattice. A number of other important properties, many of which lead to interesting special classes of lattices, will be introduced below. Lattice order - Completeness. A highly relevant class of lattices are the complete lattices. A lattice is complete if all of its subsets have both a join and a meet, which should be contrasted to the above definition of a lattice where one only requires the existence of all (non-empty) finite joins and me ...

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Lattice order, Lattice order - Formal definition, Lattice order - Lattices as posets, Lattice order - Lattices as algebraic structures, Lattice order - Connection between the two definitions, Lattice order - Examples, Lattice order - Morphisms of lattices, Lattice order - Properties of lattices, Lattice order - Completeness, Lattice order - Distributivity, Lattice order - Modularity, Lattice order - Continuity and algebraicity, Lattice order - Complements and pseudo-complements, Lattice order - Free lattices, Lattice order - Important lattice-theoretic notions

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Using the standard definition of universal algebra, a free lattice over a generating set S is a lattice L together with a function i:S→L, such that any function f from S to the underlying set of some lattice M can be factored uniquely through a lattice homomorphism f° from L to M. Stated differently, for every element s of S we find that f(s) = f°(i(s)) and that f° is the only lattice homomorphism w ...

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Lattice order, Lattice order - Formal definition, Lattice order - Lattices as posets, Lattice order - Lattices as algebraic structures, Lattice order - Connection between the two definitions, Lattice order - Examples, Lattice order - Morphisms of lattices, Lattice order - Properties of lattices, Lattice order - Completeness, Lattice order - Distributivity, Lattice order - Modularity, Lattice order - Continuity and algebraicity, Lattice order - Complements and pseudo-complements, Lattice order - Free lattices, Lattice order - Important lattice-theoretic notions

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In the following, let L be a lattice. We define some order-theoretic notions that are of particular importance in lattice theory. An element x of L is called join-irreducible iff x = a v b implies x = a or x = b for any a, b in L, if L has a 0, x is sometimes required to be different from 0. When the first condition is generalized to arbitrary joins Vai, x i ...

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Lattice order, Lattice order - Formal definition, Lattice order - Lattices as posets, Lattice order - Lattices as algebraic structures, Lattice order - Connection between the two definitions, Lattice order - Examples, Lattice order - Morphisms of lattices, Lattice order - Properties of lattices, Lattice order - Completeness, Lattice order - Distributivity, Lattice order - Modularity, Lattice order - Continuity and algebraicity, Lattice order - Complements and pseudo-complements, Lattice order - Free lattices, Lattice order - Important lattice-theoretic notions

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A semigroup with an identity element is called a monoid. Any semigroup S may be turned into a monoid simply by adjoining an element e not in S and defining es = s = se for all s ∈ S ∪ {e}. Some examples of semigroups: The positive integers with addition. Any monoid, and therefore any group. Any ideal of a ring, with the operation of multiplication. (Thus, any ring, including the integers, rational, real, complex or quaternionic numbers, function ...

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Semigroup, Semigroup - Examples of semigroups, Semigroup - Structure of semigroups

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A partition of a set X is a set of nonempty subsets of X such that every element x in X is in exactly one of these subsets. Equivalently, a set P of subsets of X, is a partition of X if No element of P is empty. (NB - some definitions do not require this) The union of the elements of P is equal to X. (We say the elements of P cover X.) The intersection of any two elements of P is empty. (We say the elements of P are pairwise disjoint.) The elements of P are som ...

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Partition of a set, Partition of a set - Definition, Partition of a set - Examples, Partition of a set - Partitions and equivalence relations, Partition of a set - Partial ordering of the lattice of partitions, Partition of a set - Noncrossing partitions, Partition of a set - The number of partitions

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The most common operation on a sieve is pullback. Pulling back a sieve S on c by an arrow f:c′→c gives a new sieve f*S on c′. This new sieve consists of all the arrows in S which factor through c′. There are several equivalent ways of defining f*S. The simplest is: For any object d of C, f*S(d) = { gSee also:

Sieve category theory, Sieve category theory - Definition, Sieve category theory - Pullback of sieves, Sieve category theory - Properties of sieves

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There is a notion of lim sup and lim inf for real-valued functions defined on a metric space whose relationship to limits of real-valued functions mirrors that of the relation between the lim sup, lim inf, and the limit of a real sequence. Given a metric space X, a subspace E contained in X, and f : E → R we define, for a any point in the closure of E: and where B(a,ε) denotes the metric ball of radius ε abou ...

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Limit superior and limit inferior, Limit superior and limit inferior - Sequences of real numbers, Limit superior and limit inferior - Functions from metric and topological spaces to the real numbers, Limit superior and limit inferior - Sequences of sets

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If a is a member of S then the set is a Dedekind cut we could call ( −∞, a ); by identifying a with it, the linearly ordered set S is embedded in the set of all Dedekind cuts of S. If the linearly ordered set S does not enjoy the least-upper-bound property, then the set of Dedekind cuts will be strictly bigger than S; conversely, if S has the least-upper-bound property, the set of its Dedekind cuts is order isomorphic to S, by identifying each ...

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Dedekind cut, Dedekind cut - Handling Dedekind cuts, Dedekind cut - Ordering Dedekind cuts, Dedekind cut - The cut construction of the real numbers, Dedekind cut - Additional structure on the cuts, Dedekind cut - Generalization: Dedekind completions in posets, Dedekind cut - Another generalization: surreal numbers

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Extended real number line - Limits. We often wish to describe the behavior of a function f(x), as either the argument x or the function value f(x) get "very big" in some sense. For example, consider the function The graph of this function has a horizontal asymptote of y = 0. Geometrically, as we move farther and farther to the right down the x-axis, the value of 1 / x2 gets closer and ...

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Extended real number line, Extended real number line - Motivation, Extended real number line - Limits, Extended real number line - Measure and integration, Extended real number line - Order and topological properties, Extended real number line - Arithmetic operations, Extended real number line - Algebraic properties, Extended real number line - Miscellaneous

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Complete Heyting algebras, frames, and locales describe entirely the same concept. Yet, subtle differences in the treatment of the three notions arise when considering them in the framework of category theory (or, on a smaller scale, abstract algebra). These differences originate in the way morphisms are defined for the structures. Frames are usually defined by the infinite distributivity law given above. Hence it is natural that a frame homomorphism is a (necessarily monotone) function that preserves finite meets and arbitrary ...

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Complete Heyting algebra, Complete Heyting algebra - Definition, Complete Heyting algebra - Differences between the concepts, Complete Heyting algebra - Literature

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Given a closure operator C, a closed element of P is an element x that is a fixed point of C, or equivalently, that is in the image of C. If a is closed and x is arbitrary, then we have x ≤ a if and only if C(x) ≤ a. So C(x) is the smallest closed element that's greater than or equal to x. We see tha ...

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Closure operator, Closure operator - Examples, Closure operator - Closed elements; properties

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Let τ1 and τ2 be two topologies on a set X such that τ1 is contained in τ2: . That is, every set open under τ1 is also open under τ2. Then the topology τ1 is said to be a coarser (weaker or smaller) topology than τ2, and τ2 is said to be a finer (stronger or larger) topology than τ1. If additionally we say τ1 is strictly coarser than τ2 and τSee also:

Comparison of topologies, Comparison of topologies - Definition, Comparison of topologies - Examples, Comparison of topologies - Properties, Comparison of topologies - Lattice of topologies

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Order of a group. Order of a group (G,*) is the cardinality (i.e. number of elements) of G. A group with finite order is called a finite group. Order of an element of a group. Suppose x∈G and there exists a positive integer m such that xm = e, then the smallest possible m is called the order of x. The order of a finite group is divisible by the order of every element. Subgroup. A subset H of a group (G,*) which remains a group when the operation * is r ...

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Glossary of group theory, Glossary of group theory - Basic definitions, Glossary of group theory - Types of groups

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Given a subgroup H and some a in G, we define the left coset aH = {ah : h in H}. Because a is invertible, the map given by is a bijection. Furthermore, every element of G is contained in precisely one left coset of H; the left cosets are the equivalence classes corresponding to the equivalence relation a1 ~ a2 iff a1−1a2 is in H. The number of left cosets of H is called the index of H in G and is denoted by [G : H ...

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Subgroup, Subgroup - Basic properties of subgroups, Subgroup - Example, Subgroup - Cosets and Lagrange's theorem

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