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The smallest (positive) [[Natural number|natural number]]: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095470/u0954701.png" />. Multiplication of any number by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095470/u0954702.png" /> produces the same number again.
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The smallest (positive) [[natural number]]: $1$. Multiplication of any number by $1$ produces the same number again.
  
An element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095470/u0954703.png" /> in a set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095470/u0954704.png" /> is called a left (right) unit (left (right) identity) with respect to a binary operation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095470/u0954705.png" /> defined on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095470/u0954706.png" /> if for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095470/u0954707.png" /> the following equation holds:
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An element $e$ in a set $M$ is called a left (right) unit (left (right) identity) with respect to a [[binary operation]] $*$ defined on $M$ if for any $a \in M$ the following equation holds:
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$$
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e * a = a \ \ \ (\ a * e = a\ ) \ .
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$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095470/u0954708.png" /></td> </tr></table>
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If there exists at least one left unit and at least one right unit, then they coincide and there are no other units. If more than one binary operation is defined on $M$ (for example, addition and multiplication in a ring), then the term unit is used for only one of these operations, usually multiplication. The unit with respect to addition is called the [[zero]] element.
  
If there exists at least one left unit and at least one right unit, then they coincide and there are no other units. If more than one binary operation is defined on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095470/u0954709.png" /> (for example, addition and multiplication in a ring), then the term unit is used for only one of these operations, usually multiplication. The unit with respect to addition is called the zero element.
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The unit of a [[lattice]] is its greatest element, that is, the identity with respect to the operation of intersection.
  
The unit of a lattice is its greatest element, that is, the identity with respect to the operation of intersection (cf. also [[Lattice|Lattice]]).
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In an [[integral domain]] $D$, any invertible element, that is, an element $u$ that has an inverse $u^{-1}$ such that $u.u^{-1} = u^{-1}.u = 1$, is called a unit, or a divisor of unity. The units of an integral domain form a group under multiplication. The same terminology is sometimes preserved when passing to the field of fractions of the ring $D$ (that is, the units of $D$ itself are called units of the field of fractions). For example, the units of an algebraic number field $k$ are the units of the ring of algebraic integers of $k$, the $p$-adic units are the units of the ring of [[P-adic number|$p$-adic numbers]], etc.
  
In an integral domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095470/u09547010.png" />, any invertible element, that is, an element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095470/u09547011.png" /> that has an inverse <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095470/u09547012.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095470/u09547013.png" />, is called a unit, or a divisor of unity. The units of an integral domain form a group under multiplication. The same terminology is sometimes preserved when passing to the field of fractions of the ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095470/u09547014.png" /> (that is, the units of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095470/u09547015.png" /> itself are called units of the field of fractions). For example, the units of an algebraic number field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095470/u09547016.png" /> are the units of the ring of algebraic integers of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095470/u09547017.png" />, the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095470/u09547018.png" />-adic units are the units of the ring of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095470/u09547019.png" />-adic numbers (cf. [[P-adic number|<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095470/u09547020.png" />-adic number]]), etc.
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The unit morphism (identity morphism) of an object $X$ in a [[category]] is the (unique) morphism $1_X : X \rightarrow X$ satisfying $1_X f = f$ and $g 1_X = g$ for all $f : Y \rightarrow X$ and all $g : X \rightarrow Z$.
 
 
The unit morphism (identity morphism) of an object <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095470/u09547021.png" /> in a [[Category|category]] is the (unique) morphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095470/u09547022.png" /> satisfying <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095470/u09547023.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095470/u09547024.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095470/u09547025.png" /> and all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095470/u09547026.png" />.
 
  
  
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The Russian language uses the same word to translate the English words  "unit"  and  "identity" . In English, the word  "identity"  is more commonly used than  "unit"  in senses 2) and 5) above, but  "unit"  is commoner in the other three senses.
 
The Russian language uses the same word to translate the English words  "unit"  and  "identity" . In English, the word  "identity"  is more commonly used than  "unit"  in senses 2) and 5) above, but  "unit"  is commoner in the other three senses.
  
A standard length when measuring things or comparing objects is called a unit of length, or simply a unit. More generally, various somethings one of whose main parameters is equal to the unit length is called a  "unit something" ; e.g. unit vector, unit circle, unit disc, unit cube, unit <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095470/u09547028.png" />-cube, unit sphere, unit ball, unit <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095470/u09547030.png" />-ball (unit cell), etc.
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A standard length when measuring things or comparing objects is called a unit of length, or simply a unit. More generally, various somethings one of whose main parameters is equal to the unit length is called a  "unit something" ; e.g. unit vector, unit circle, unit disc, unit cube, unit $n$-cube, unit sphere, unit ball, unit $n$-ball (unit cell), etc.
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Latest revision as of 19:25, 3 April 2016

The smallest (positive) natural number: $1$. Multiplication of any number by $1$ produces the same number again.

An element $e$ in a set $M$ is called a left (right) unit (left (right) identity) with respect to a binary operation $*$ defined on $M$ if for any $a \in M$ the following equation holds: $$ e * a = a \ \ \ (\ a * e = a\ ) \ . $$

If there exists at least one left unit and at least one right unit, then they coincide and there are no other units. If more than one binary operation is defined on $M$ (for example, addition and multiplication in a ring), then the term unit is used for only one of these operations, usually multiplication. The unit with respect to addition is called the zero element.

The unit of a lattice is its greatest element, that is, the identity with respect to the operation of intersection.

In an integral domain $D$, any invertible element, that is, an element $u$ that has an inverse $u^{-1}$ such that $u.u^{-1} = u^{-1}.u = 1$, is called a unit, or a divisor of unity. The units of an integral domain form a group under multiplication. The same terminology is sometimes preserved when passing to the field of fractions of the ring $D$ (that is, the units of $D$ itself are called units of the field of fractions). For example, the units of an algebraic number field $k$ are the units of the ring of algebraic integers of $k$, the $p$-adic units are the units of the ring of $p$-adic numbers, etc.

The unit morphism (identity morphism) of an object $X$ in a category is the (unique) morphism $1_X : X \rightarrow X$ satisfying $1_X f = f$ and $g 1_X = g$ for all $f : Y \rightarrow X$ and all $g : X \rightarrow Z$.


Comments

The Russian language uses the same word to translate the English words "unit" and "identity" . In English, the word "identity" is more commonly used than "unit" in senses 2) and 5) above, but "unit" is commoner in the other three senses.

A standard length when measuring things or comparing objects is called a unit of length, or simply a unit. More generally, various somethings one of whose main parameters is equal to the unit length is called a "unit something" ; e.g. unit vector, unit circle, unit disc, unit cube, unit $n$-cube, unit sphere, unit ball, unit $n$-ball (unit cell), etc.

How to Cite This Entry:
Unit. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Unit&oldid=38539
This article was adapted from an original article by O.A. Ivanov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article