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Isomorphism




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In mathematics (particularly in abstract algebra), two mathematical structures are isomorphic when they are the same in some sense. More specifically, an isomorphism is a function between two structures that preserves the relationships between the parts (see https://en.wikipedia.org/wiki/Isomorphism#Examples). To indicate isomorphism between two structures [math]\mathcal{A}[/math] and [math]\mathcal{B}[/math], one often writes [math]\mathcal{A} \cong \mathcal{B}[/math].[1][2]

Using the language of category theory, this means that morphisms map to morphisms without breaking composition. An isomorphism is also a homomorphism that is one-to-one.[3]

As an example, one can consider the operation of adding integers Z. The doubling function φ(x) = 2x maps elements of Z to elements of the even integers 2Z. Since φ(a+b) = 2(a+b) = 2a+2b = φ(a)+φ(b), adding in Z is structurally identical as adding in 2Z (which makes this an example of isomorphism).

Related pages

References

  1. "Comprehensive List of Algebra Symbols" (in en-US). 2020-03-25. https://mathvault.ca/hub/higher-math/math-symbols/algebra-symbols/. 
  2. Weisstein, Eric W.. "Isomorphism" (in en). https://mathworld.wolfram.com/Isomorphism.html. 
  3. "Definition of ISOMORPHISM" (in en). https://www.merriam-webster.com/dictionary/isomorphism.