View source for Isomorphism

{{TEX|done}}
{{MSC|18-01,|00A05,08-01,18Axx,18Dxx}}

An isomorphism is 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 $\def\phi{\varphi}\phi$ 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|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|homomorphism]] that is a
[[Bijection|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|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|logarithmic function]] (cf. also
[[Exponential function, real|Exponential function, real]]).

====References====
{| 
|- 
|valign="top"|{{Ref|Ad}}||valign="top"|J. Adámek, "Theory of mathematical structures", Reidel (1983) {{MR|0735079}} {{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}}
|- 
|}