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A correspondence (relation) between objects or systems of objects expressing the equality of their structures in some sense. An isomorphism in an arbitrary [[Category|category]] is an invertible [[Morphism|morphism]], that is, a morphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052840/i0528401.png" /> for which there exists a morphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052840/i0528402.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052840/i0528403.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052840/i0528404.png" /> are both identity morphisms.
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{{TEX|done}
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{{MSC|18-01,|00A05,08-01,18Axx,18Dxx}}
  
The concept of an isomorphism arose in connection with concrete algebraic systems (initially, with groups) and was extended in a natural way to wider classes of mathematical structures. A classical example of isomorphic, "identically constructed" , systems is the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052840/i0528405.png" /> of real numbers with the operation of addition and the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052840/i0528406.png" /> of positive real numbers with the operation of multiplication.
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A correspondence (relation) between objects or systems of objects
 +
expressing the equality of their structures in some sense. An
 +
isomorphism in an arbitrary
 +
[[Category|category]] is an invertible
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[[Morphism|morphism]], that is, a morphism $\def\phi{\varphi}\phi$ for which there exists
 +
a morphism $\phi^{-1}$ such that $\phi^{-1}\phi$ and $\phi\phi^{-1}$ are both identity morphisms.
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052840/i0528407.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052840/i0528408.png" /> be algebraic systems (cf. [[Algebraic system|Algebraic system]]) of the same type, written in the signature
+
The concept of an isomorphism arose in connection with concrete
 +
algebraic systems (initially, with groups) and was extended in a
 +
natural way to wider classes of mathematical structures. A classical
 +
example of isomorphic, "identically constructed" , systems is the set
 +
$\R$ of real numbers with the operation of addition and the set $P$ of
 +
positive real numbers with the operation of multiplication.
  
<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/i/i052/i052840/i0528409.png" /></td> </tr></table>
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Let $\mathfrak{A}$ and $\mathfrak{A'}$ be algebraic systems (cf.
 +
[[Algebraic system|Algebraic system]]) of the same type, written in
 +
the signature
 +
$$\{F_i: i\in I\}\cup\{P_j: j\in J\}$$
 +
with function symbols $F_i$, $i\in I$, and predicate
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symbols $P_j$, $j\in J$:
  
with function symbols <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052840/i05284010.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052840/i05284011.png" />, and predicate symbols <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052840/i05284012.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052840/i05284013.png" />:
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$$ A = \langle A; \{F_i: i\in I\},\; \{P_j: j\in J\}\rangle,$$
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$$ A' = \langle A'; \{F_i: i\in I\},\; \{P_j: j\in J\}\rangle.$$
  
<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/i/i052/i052840/i05284014.png" /></td> </tr></table>
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An
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isomorphism, or isomorphic mapping, from $A$ onto $A'$ is a one-to-one
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mapping $\phi$ from the set $A$ onto the set $A'$ with the properties
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$$\def\a{\alpha} \phi(F_i(\a_1,\dots,\a_{n_i})) = F_i(\phi(\a_1),\dots,\phi(\a_{n_i})),$$
  
<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/i/i052/i052840/i05284015.png" /></td> </tr></table>
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$$P_j(\a_1,\dots,\a_{m_j})\Leftrightarrow P_j(\phi(\a_1),\dots,\phi(\a_{m_j}))$$
  
An isomorphism, or isomorphic mapping, from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052840/i05284016.png" /> onto <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052840/i05284017.png" /> is a one-to-one mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052840/i05284018.png" /> from the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052840/i05284019.png" /> onto the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052840/i05284020.png" /> with the properties:
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for all $\a_1,\a_2,\dots$ in $A$ and all $i\in I$, $j\in J$. Thus, in every category of
 +
algebraic systems, an isomorphism is a
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[[Homomorphism|homomorphism]] that is a
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[[Bijection|bijection]]. An isomorphism of an algebraic system onto
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itself is called an automorphism.
  
<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/i/i052/i052840/i05284021.png" /></td> </tr></table>
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The relation of isomorphism is reflexive, symmetric and transitive,
 
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that is, it is an equivalence relation splitting any set on which it
<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/i/i052/i052840/i05284022.png" /></td> </tr></table>
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is defined into disjoint equivalence classes — the classes of
 
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pairwise-isomorphic systems. A class of algebraic systems which is a
for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052840/i05284023.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052840/i05284024.png" /> and all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052840/i05284025.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052840/i05284026.png" />. Thus, in every category of algebraic systems, an isomorphism is a [[Homomorphism|homomorphism]] that is a [[Bijection|bijection]]. An isomorphism of an algebraic system onto itself is called an automorphism.
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union of such classes is called an abstract class (cf.
 
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[[Algebraic systems, class of|Algebraic systems, class of]]).
The relation of isomorphism is reflexive, symmetric and transitive, that is, it is an equivalence relation splitting any set on which it is defined into disjoint equivalence classes — the classes of pairwise-isomorphic systems. A class of algebraic systems which is a union of such classes is called an abstract class (cf. [[Algebraic systems, class of|Algebraic systems, class of]]).
 
  
  
  
 
====Comments====
 
====Comments====
The isomorphism between <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052840/i05284027.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052840/i05284028.png" /> mentioned in the main article above can be explicitly given by the means of the exponential mapping or its inverse, the [[Logarithmic function|logarithmic function]] (cf. also [[Exponential function, real|Exponential function, real]]).
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The isomorphism between $\R$ and $P$ mentioned in the
 +
main article above can be explicitly given by the means of the
 +
exponential mapping or its inverse, the
 +
[[Logarithmic function|logarithmic function]] (cf. also
 +
[[Exponential function, real|Exponential function, real]]).
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  P.M. Cohn,   "Universal algebra" , Reidel (1981)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  J. Adámek,   "Theory of mathematical structures" , Reidel (1983)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"B. Mitchell,   "Theory of categories" , Acad. Press (1965) pp. 7</TD></TR></table>
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{|
 +
|-
 +
|valign="top"|{{Ref|Ad}}||valign="top"|J. Adámek, "Theory of mathematical structures", Reidel (1983) {{M|R0735079}} {{ZBL|0523.18001}}
 +
|-
 +
|valign="top"|{{Ref|Co}}||valign="top"| P.M. Cohn, "Universal algebra", Reidel (1981) {{MR|0620952}} {{ZBL|0461.08001}}
 +
|-
 +
|valign="top"|{{Ref|Mi}}||valign="top"|B. Mitchell, "Theory of categories", Acad. Press (1965) pp. 7 {{MR|0202787}} {{ZBL|0136.00604}}
 +
|-
 +
|}

Revision as of 11:42, 18 February 2012

{{TEX|done} 2010 Mathematics Subject Classification: Primary: 18-01, Secondary: 00A0508-0118Axx18Dxx [MSN][ZBL]

A correspondence (relation) between objects or systems of objects expressing the equality of their structures in some sense. An isomorphism in an arbitrary category is an invertible morphism, that is, a morphism $\def\phi{\varphi}\phi$ for which there exists a morphism $\phi^{-1}$ such that $\phi^{-1}\phi$ and $\phi\phi^{-1}$ are both identity morphisms.

The concept of an isomorphism arose in connection with concrete algebraic systems (initially, with groups) and was extended in a natural way to wider classes of mathematical structures. A classical example of isomorphic, "identically constructed" , systems is the set $\R$ of real numbers with the operation of addition and the set $P$ of positive real numbers with the operation of multiplication.

Let $\mathfrak{A}$ and $\mathfrak{A'}$ be algebraic systems (cf. Algebraic system) of the same type, written in the signature $$\{F_i: i\in I\}\cup\{P_j: j\in J\}$$ with function symbols $F_i$, $i\in I$, and predicate symbols $P_j$, $j\in J$:

$$ A = \langle A; \{F_i: i\in I\},\; \{P_j: j\in J\}\rangle,$$ $$ A' = \langle A'; \{F_i: i\in I\},\; \{P_j: j\in J\}\rangle.$$

An isomorphism, or isomorphic mapping, from $A$ onto $A'$ is a one-to-one mapping $\phi$ from the set $A$ onto the set $A'$ with the properties $$\def\a{\alpha} \phi(F_i(\a_1,\dots,\a_{n_i})) = F_i(\phi(\a_1),\dots,\phi(\a_{n_i})),$$

$$P_j(\a_1,\dots,\a_{m_j})\Leftrightarrow P_j(\phi(\a_1),\dots,\phi(\a_{m_j}))$$

for all $\a_1,\a_2,\dots$ in $A$ and all $i\in I$, $j\in J$. Thus, in every category of algebraic systems, an isomorphism is a homomorphism that is a bijection. An isomorphism of an algebraic system onto itself is called an automorphism.

The relation of isomorphism is reflexive, symmetric and transitive, that is, it is an equivalence relation splitting any set on which it is defined into disjoint equivalence classes — the classes of pairwise-isomorphic systems. A class of algebraic systems which is a union of such classes is called an abstract class (cf. Algebraic systems, class of).


Comments

The isomorphism between $\R$ and $P$ mentioned in the main article above can be explicitly given by the means of the exponential mapping or its inverse, the logarithmic function (cf. also Exponential function, real).

References

[Ad] J. Adámek, "Theory of mathematical structures", Reidel (1983) Template:M Zbl 0523.18001
[Co] P.M. Cohn, "Universal algebra", Reidel (1981) MR0620952 Zbl 0461.08001
[Mi] B. Mitchell, "Theory of categories", Acad. Press (1965) pp. 7 MR0202787 Zbl 0136.00604
How to Cite This Entry:
Isomorphism. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Isomorphism&oldid=12359
This article was adapted from an original article by O.A. IvanovaD.M. Smirnov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article