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medial

 
Dictionary: me·di·al   (mē'dē-əl) pronunciation
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adj.
  1. Relating to, situated in, or extending toward the middle; median.
  2. Linguistics. Being a sound, syllable, or letter occurring between the initial and final positions in a word or morpheme.
  3. Mathematics. Being or relating to an average or a mean.
  4. Average; ordinary.
n. Linguistics
  1. A voiced stop, such as (b), (d), or (g). Also called media.
  2. A sound, letter, or form of a letter that is neither initial nor final.

[Late Latin mediālis, from Latin medius, middle.]

medially me'di·al·ly adv.

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Thesaurus: medial
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adjective

  1. At, in, near, or being the center: center, central, median, mid, middle. See edge/center.
  2. Not extreme: central, intermediate, mean3, median, mid, middle, middle-of-the-road, midway. See edge/center.

Dental Dictionary: medial
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(me'de-əl)
adj

Located in or directed toward the middle; closer to the body’s midline.

Sports Science and Medicine: medial
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Toward the midline of the body.

Veterinary Dictionary: medial
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Pertaining to or situated toward the midline.

  • m. nasal process — one of the frontal processes derived from frontonasal mesenchyme and forming part of the border of the nasal pits, the future nostrils.
  • m. palatine process — see palatine process.
  • m. patellar ligament — in the species in which the tendon is trifurcated (horse, cattle), the largest and most medial of the three patellar ligaments.
Wikipedia: Medial
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This article is about medial in mathematics. For other uses, see medial (disambiguation).

Contents

Medial magmas

In abstract algebra, a medial magma (or medial groupoid) is a set with a binary operation which satisfies the identity

(x \cdot y) \cdot (u \cdot v) = (x \cdot u) \cdot (y \cdot v), or more simply, xy\cdot uv = xu\cdot yv

using the convention that juxtaposition has higher precedence. This identity has been variously called medial, abelian, alternation, transposition, bi-commutative, bisymmetric, surcommutative, entropic, etc.[1]

Any commutative semigroup is a medial magma, and a medial magma has an identity element if and only if it is a commutative monoid. An elementary example of a nonassociative medial quasigroup can be constructed as follows: take an abelian group except the group of order 2 (written additively) and define a new operation by x * y = (− x) + (− y).

A magma M is medial if and only if its binary operation is a homomorphism from the Cartesian square M x M to M. This can easily be expressed in terms of a commutative diagram, and thus leads to the notion of a medial magma object in a category with a cartesian product. (See the discussion in auto magma object.)

If f and g are endomorphisms of a medial magma, then the mapping f.g defined by pointwise multiplication

(f\cdot g)(x) = f(x)\cdot g(x)

is itself an endomorphism.

Generalizations

The term medial or (more commonly) entropic is also used for a generalization to multiple operations. An algebraic structure is an entropic algebra[2] if every two operations satisfy a generalization of the medial identity. Let f and g be operations of arity m and n, respectively. Then f and g are required to satisfy

f( g(x_{11}, \ldots, x_{1n}), \ldots, g(x_{m1}, \ldots, g_{mn}) ) = g( f(x_{11}, \ldots, x_{m1}), \ldots, f(x_{1n}, \ldots, f_{mn}) ).

See also

External links

  1. ^ Historical comments J.Jezek and T.Kepka: Medial groupoids Rozpravy CSAV, Rada mat. a prir. ved 93/2 (1983), 93 pp
  2. ^ [1]

This entry is from Wikipedia, the leading user-contributed encyclopedia. It may not have been reviewed by professional editors (see full disclaimer)

Translations: Medial
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Dansk (Danish)
adj. - midt-, midter-, i midten, middelstor
n. - medial (fonetisk)

Nederlands (Dutch)
middel-, midden-, gemiddeld, mediaal

Français (French)
adj. - (Ling) médial, médian (position), (Math) moyen
n. - (Phon) médiale

Deutsch (German)
adj. - mittler, medial
n. - Media

Ελληνική (Greek)
adj. - διάμεσος, μεσαίος, μέσος
n. - μεσαίο γράμμα

Italiano (Italian)
medio, mediano

Português (Portuguese)
adj. - mediano
n. - letra média (f) (Gram.)

Русский (Russian)
средний, медиальный

Español (Spanish)
adj. - central, medial
n. - letra del medio, letra interna

Svenska (Swedish)
adj. - medial, mitt-, genomsnitts-
n. - inljud

中文（简体）(Chinese (Simplified))
中间的, 普通的, 平均的, 中间字母

中文（繁體）(Chinese (Traditional))
adj. - 中間的, 普通的, 平均的
n. - 中間字母

한국어 (Korean)
adj. - 중앙의, 일반적인, 기준의
n. - 중간, 평균

日本語 (Japanese)
adj. - 中間の, 平均の, 中央の, 語中の

العربيه (Arabic)
‏(صفه) توسطي (الاسم) متوسط‏

עברית (Hebrew)
adj. - ‮ממוצע, תיכון, אמצעי‬
n. - ‮ממוצע, צליל אמצעי (פונטיקה)‬

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Dictionary. The American Heritage® Dictionary of the English Language, Fourth Edition Copyright © 2007, 2000 by Houghton Mifflin Company. Updated in 2009. Published by Houghton Mifflin Company. All rights reserved.  Read more
Thesaurus. Roget's II: The New Thesaurus, Third Edition by the Editors of the American Heritage® Dictionary Copyright © 1995 by Houghton Mifflin Company. Published by Houghton Mifflin Company. All rights reserved.  Read more
Dental Dictionary. Mosby's Dental Dictionary. Copyright © 2004 by Elsevier, Inc. All rights reserved.  Read more
Sports Science and Medicine. The Oxford Dictionary of Sports Science & Medicine. Copyright © Michael Kent 1998, 2006, 2007. All rights reserved.  Read more
Veterinary Dictionary. Saunders Comprehensive Veterinary Dictionary 3rd Edition. Copyright © 2007 by D.C. Blood, V.P. Studdert and C.C. Gay, Elsevier. All rights reserved.  Read more
Wikipedia. This article is licensed under the Creative Commons Attribution/Share-Alike License. It uses material from the Wikipedia article "Medial".  Read more
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