Extension, extend or extended may refer to:
Mathematics
Logic or set theory
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Axiom of extensionality
The axiom of extensionality, also called the axiom of extent, is an axiom used in many forms of axiomatic set theory, such as Zermelo–Fraenkel set theory. The axiom defines what a Set (mathematics), set is. Informally, the axiom means that the ...
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Extensible cardinal In mathematics, extendible cardinals are large cardinals introduced by , who was partly motivated by reflection principles. Intuitively, such a cardinal represents a point beyond which initial pieces of the universe of sets start to look similar, ...
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Extension (model theory)
In mathematical logic, an (induced) substructure or (induced) subalgebra is a structure whose domain is a subset of that of a bigger structure, and whose functions and relations are restricted to the substructure's domain. Some examples of subalg ...
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Extension (proof theory)
In mathematical logic, a conservative extension is a supertheory of a theory which is often convenient for proving theorems, but proves no new theorems about the language of the original theory. Similarly, a non-conservative extension is a superth ...
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Extension (predicate logic)
The extension of a predicatea truth-valued functionis the set of tuples of values that, used as arguments, satisfy the predicate. Such a set of tuples is a relation.
Examples
For example, the statement "''d2'' follows the weekday ''d1''" can ...
, the set of tuples of values that satisfy the predicate
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Extension (semantics)
In any of several fields of study that treat the use of signs — for example, in linguistics, logic, mathematics, semantics, semiotics, and philosophy of language — the extension of a concept, idea, or sign consists of the things to which it ap ...
, the set of things to which a property applies
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Extension (simplicial set)
In higher category theory in mathematics, the extension of Simplicial set, simplicial sets (extension functor or Ex functor) is an endofunctor on the category of simplicial sets. Due to many remarkable properties, the extension functor has plenty a ...
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Extension by definitions
In mathematical logic, more specifically in the proof theory of first-order theories, extensions by definitions formalize the introduction of new symbols by means of a definition. For example, it is common in naive set theory to introduce a symbo ...
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Extensional definition
In logic, extensional and intensional definitions are two key ways in which the objects, concepts, or referents a term refers to can be defined. They give meaning or denotation to a term.
An intensional definition gives meaning to a term by sp ...
, a definition that enumerates every individual a term applies to
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Extensionality
In logic, extensionality, or extensional equality, refers to principles that judge objects to be equality (mathematics), equal if they have the same external properties. It stands in contrast to the concept of intensionality, which is concerned wi ...
Other uses
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Extension of a function
In mathematics, the restriction of a function f is a new function, denoted f\vert_A or f , obtained by choosing a smaller domain A for the original function f.
The function f is then said to extend f\vert_A.
Formal definition
Let f : E \to ...
, defined on a larger domain
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Extension of a polyhedron, in geometry
* Extension of a
line segment
In geometry, a line segment is a part of a line (mathematics), straight line that is bounded by two distinct endpoints (its extreme points), and contains every Point (geometry), point on the line that is between its endpoints. It is a special c ...
(finite) into an infinite line (e.g.,
extended base
In geometry, a base is a Edge (geometry), side of a polygon or a face (geometry), face of a polyhedron, particularly one oriented perpendicular to the direction in which Height#In mathematics, height is measured, or on what is considered to be th ...
)
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Exterior algebra
In mathematics, the exterior algebra or Grassmann algebra of a vector space V is an associative algebra that contains V, which has a product, called exterior product or wedge product and denoted with \wedge, such that v\wedge v=0 for every vector ...
, Grassmann's theory of extension, in geometry
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Field extension
In mathematics, particularly in algebra, a field extension is a pair of fields K \subseteq L, such that the operations of ''K'' are those of ''L'' restricted to ''K''. In this case, ''L'' is an extension field of ''K'' and ''K'' is a subfield of ...
, in Galois theory
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Group extension
In mathematics, a group extension is a general means of describing a group in terms of a particular normal subgroup and quotient group. If Q and N are two groups, then G is an extension of Q by N if there is a short exact sequence
:1\to N\;\ove ...
, in abstract algebra and homological algebra
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Homotopy extension property
In mathematics, in the area of algebraic topology, the homotopy extension property indicates which homotopies defined on a subspace can be extended to a homotopy defined on a larger space. The homotopy extension property of cofibrations is du ...
, in topology
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Kolmogorov extension theorem
In mathematics, the Kolmogorov extension theorem (also known as Kolmogorov existence theorem, the Kolmogorov consistency theorem or the Daniell-Kolmogorov theorem) is a theorem that guarantees that a suitably "consistent" collection of finite-dim ...
, in probability theory
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Linear extension
In order theory, a branch of mathematics, a linear extension of a partial order is a total order (or linear order) that is compatible with the partial order. As a classic example, the lexicographic order of totally ordered sets is a linear extensi ...
, in order theory
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Sheaf extension, in algebraic geometry
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Tietze extension theorem
In topology, the Tietze extension theorem (also known as the Tietze– Urysohn– Brouwer extension theorem or Urysohn-Brouwer lemma) states that any real-valued, continuous function on a closed subset of a normal topological space
In mathe ...
, in topology
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Whitney extension theorem
In mathematics, in particular in mathematical analysis, the Whitney extension theorem is a partial converse to Taylor's theorem. Roughly speaking, the theorem asserts that if ''A'' is a closed subset of a Euclidean space, then it is possible to ...
, in differential geometry
Music
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Extension (music)
In music, an extension is a set of musical notes that lie outside the standard range or tessitura.
Staff
A note that lies outside the lines of a musical staff is an extension of the staff. The note will lie on a ledger line. Middle C, f ...
, notes that fit outside the standard range
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''Extended'' (Solar Fields album), 2005
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''Extension'' (George Braith album), 1964
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''Extension'' (Clare Fischer album), 1963
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''Extensions'' (Ahmad Jamal album), 1965
* ''Extensions'', a 1969 album by
Mystic Moods Orchestra
The Mystic Moods Orchestra was a group known for mixing orchestral pop, environmental sounds, and pioneering recording techniques. It was created by audiophile Brad Miller. The first Mystic Moods Orchestra album, ''One Stormy Night'', was released ...
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''Extensions'' (McCoy Tyner album), 1970
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''Extensions'' (Dave Holland album), 1988
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''Extensions'' (The Manhattan Transfer album), 1979
* Extension (The Extended Mixes), 2023 album reissue by
Kylie Minogue
Kylie Ann Minogue (; born 28 May 1968) is an Australian singer, songwriter, and actress. Frequently referred to as the "Honorific nicknames in popular music, Princess of Pop", she has achieved recognition in both the music industry and fas ...
Places
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Extension, British Columbia
Extension is an unincorporated community near the east coast of southern Vancouver Island, British Columbia. The location off BC Highway 19 is about by road northwest of Victoria, and south of Nanaimo.
Discovery
Louis Stark, an African America ...
, a village near Regional District of Nanaimo, British Columbia, Canada
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Extension, Louisiana
Extension is an Unincorporated area#United States, unincorporated community in Franklin Parish, Louisiana, Franklin Parish, Louisiana, United States. Its ZIP code is 71243.
Notable people
Singer and songwriter Richard Berry (musician), Richard Be ...
, unincorporated community, United States
Science
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Extension (geology)
Extensional tectonics is concerned with the structures formed by, and the Tectonics, tectonic processes associated with, the stretching of a planetary body's Crust (geology), crust or lithosphere.
Deformation styles
The types of structure and the ...
, relating to the pulling apart of the Earth's crust and lithosphere
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Extension (anatomy), a movement at a
joint
A joint or articulation (or articular surface) is the connection made between bones, ossicles, or other hard structures in the body which link an animal's skeletal system into a functional whole.Saladin, Ken. Anatomy & Physiology. 7th ed. McGraw- ...
that increases the angle between the two ends of the joint; the opposite of
flexion
Motion, the process of movement, is described using specific anatomical terminology, anatomical terms. Motion includes movement of Organ (anatomy), organs, joints, Limb (anatomy), limbs, and specific sections of the body. The terminology used de ...
* Extension locus, the gene locus of
Melanocortin 1 receptor
The melanocortin 1 receptor (MC1R), also known as melanocyte-stimulating hormone receptor (MSHR), melanin-activating peptide receptor, or melanotropin receptor, is a G protein–coupled receptor that binds to a class of pituitary peptide hormon ...
Other uses
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Agricultural Extension
Agricultural extension is the application of scientific research and new knowledge to agricultural practices through farmer education. The field of 'extension' now encompasses a wider range of communication and learning activities organized for r ...
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Browser extension
A browser extension is a software module for customizing a web browser. Browsers typically allow users to install a variety of extensions, including user interface modifications, cookie management, ad blocking, and the custom scripting and st ...
, a small software module for customizing a web browser
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Building extension
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Continuing education
Continuing education is the education undertaken after initial education for either personal or professional reasons. The term is used mainly in the United States and Canada.
Recognized forms of post-secondary learning activities within the d ...
, or extension school, a school for continuing education
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Cooperative State Research, Education, and Extension Service
The Cooperative State Research, Education, and Extension Service (CSREES) was an Extension agency within the U.S. Department of Agriculture (USDA), part of the executive branch of the federal government. The 1994 Department Reorganization Act ...
, a former division of the U.S. Department of Agriculture
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Extension (metaphysics) In metaphysics, extension signifies both 'stretching out' (Latin: ''extensio'') as well as later 'taking up space', and most recently, spreading one's internal mental cognition into the external world.
The history of thinking about ''extension'' ca ...
, the property of stretching out or taking up space
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Extension (telephone)
In residential telephony, an extension telephone is an additional telephone wired to the same telephone line as another. In middle 20th century telephone jargon, the first telephone on a line was a "Main Station" and subsequent ones "Extensions" ...
, telephone line attached to a main line or to a PBX or Centrex system
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Extension cord
An extension cord (US), extension cable, power extender, drop cord, or extension lead (UK) is a length of flexible electrical power cable (flex) with a AC power plugs and sockets, plug on one end and one or more sockets on the other end (usuall ...
, power cable with a plug on one end and one or more sockets on the other end
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Eyelash extensions, material applied to eyelashes
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Hair extensions
Artificial hair integrations, more commonly known as hair extensions, hair weaves, or fake hair, are cosmetic treatments which add length to human hair. Hair extensions are usually clipped, glued, or sewn on natural hair by incorporating addition ...
, strands of hair added to existing hair
* Extension, the building of community capacity by outsiders, for instance
agricultural extension
Agricultural extension is the application of scientific research and new knowledge to agricultural practices through farmer education. The field of 'extension' now encompasses a wider range of communication and learning activities organized for r ...
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Filename extension
A filename extension, file name extension or file extension is a suffix to the name of a computer file (for example, .txt, .mp3, .exe) that indicates a characteristic of the file contents or its intended use. A filename extension is typically d ...
, for computer file systems
See also
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Extensive (disambiguation)
{{Disambiguation