medial

American Heritage Dictionary:

me·di·al

(mē'dē-əl) pronunciation

adj.

Relating to, situated in, or extending toward the middle; median.
Linguistics. Being a sound, syllable, or letter occurring between the initial and final positions in a word or morpheme.
Mathematics. Being or relating to an average or a mean.
Average; ordinary.

n. Linguistics

A voiced stop, such as (b), (d), or (g). Also called media.
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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medial

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adjective

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

Oxford Dictionary of Sports Science & Medicine:

medial

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Toward the midline of the body.

Oxford Dictionary of Biochemistry:

medial

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of, situated in, or towards the middle.
of, or pertaining to, a media (def. 2).

Previous:	media, mechanotransduction, mechanochemical coupling
Next:	median, median effective dose, median infectious dose

Saunders 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.

Mosby's Dental Dictionary:

medial

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(me'de-əl)
adj

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

Random House Word Menu:

categories related to 'medial'

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Random House Word Menu by Stephen Glazier

For a list of words related to medial, see:

Relative Positions - medial: (adj) in the middle

Rhymes:

medial

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See words rhyming with "medial."

Wikipedia on Answers.com:

Medial

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This article is about medial in mathematics. For other uses, see medial (disambiguation).

Contents

1 Medial magmas
2 Bruck-Toyoda theorem
3 Generalizations
4 See also
5 References

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, interchange, 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. Another class of semigroups forming medial magmas are the normal bands.^[2] Medial magmas need not be associative: for any nontrivial abelian group and integers m ≠ n, replacing the group operation $x + y$ with the binary operation $x \cdot y = mx+ny$ yields a medial magma which in general is neither associative nor commutative.

A magma M is medial if and only if its binary operation is a homomorphism from the Cartesian square M×M to M.^{[clarification needed]} 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.

Bruck-Toyoda theorem

The Bruck-Toyoda theorem provides the following characterization of medial quasigroups. Given an abelian group A and two automorphisms φ and ψ of A, define an operation ∗ on A by

x ∗ y = φ(x) + ψ(y) + c

where c some fixed element of A. It is not hard to prove that A forms a medial quasigroup under this operation. The Bruck-Toyoda theorem states that every medial quasigroup is of this form, i.e. is isomorphic to a quasigroup defined from an abelian group in this way.^[3] In particular, every medial quasigroup is isotopic to an abelian group.

Generalizations

The term medial or (more commonly) entropic is also used for a generalization to multiple operations. An algebraic structure is an entropic algebra^[4] 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

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

References

^ Historical comments J.Jezek and T.Kepka: Medial groupoids Rozpravy CSAV, Rada mat. a prir. ved 93/2 (1983), 93 pp
^ Yamada, Miyuki (1971), "Note on exclusive semigroups", Semigroup Forum 3 (1): 160–167, doi:10.1007/BF02572956 .
^ Kuzʹmin, E. N. and Shestakov, I. P. (1995). "Non-associative structures". Algebra VI. Encyclopaedia of Mathematical Sciences. 6. Berlin, New York: Springer-Verlag. pp. 197–280. ISBN 978-3540546993.
^ [1]

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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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		American Heritage 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
		Roget's Thesaurus. Roget's II: The New Thesaurus, Third Edition by the Editors of the American Heritage® Dictionary Copyright © 1995 byHoughton Mifflin Company. Published by Houghton Mifflin Company. All rights reserved. Read more
		Oxford Dictionary of Sports Science & Medicine. The Oxford Dictionary of Sports Science & Medicine. Copyright © Michael Kent 1998, 2006, 2007. All rights reserved. Read more
		Oxford Dictionary of Biochemistry. Oxford University Press. Oxford Dictionary of Biochemistry and Molecular Biology © 1997, 2000, 2006 All rights reserved. Read more
		Saunders 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
		Mosby's Dental Dictionary. Mosby's Dental Dictionary. Copyright © 2004 by Elsevier, Inc. All rights reserved. Read more
		Random House Word Menu. © 2010 Write Brothers Inc. Word Menu is a registered trademark of the Estate of Stephen Glazier. Write Brothers Inc. All rights reserved. Read more
		Rhymes. Oxford University Press. © 2006, 2007 All rights reserved. Read more
		Wikipedia on Answers.com. This article is licensed under the Creative Commons Attribution/Share-Alike License. It uses material from the Wikipedia article Medial. Read more
		Translations. Copyright © 2007, WizCom Technologies Ltd. All rights reserved. Read more

medial

me·di·al

medial

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categories related to 'medial'

medial

Medial

Medial magmas

Bruck-Toyoda theorem

Generalizations

See also

References

Medial