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  "body": "\n## Summary\n\nAn isomorphism between two [[structures|983GZBQYT4]] of the same signature is a bijection that preserves and reflects all operations, constants and relations.\nTwo structures are isomorphic when an isomorphism between them exists; then they have the same properties expressible in their signature.\n[[Group isomorphisms|08MG7H8JW5]], [[ring isomorphisms|0KN2S81K72]], [[linear isomorphisms|0B8T10CE7N]] and [[order isomorphisms|0D3WA36K2Q]] are special cases.\nAn isomorphism of a structure with itself is an [[automorphism|DSQ17TNW1C]].\n\n## Explanations\n\n### L1 human\n\nTwo structures are isomorphic when they are the same up to renaming of their elements.\nThe even integers with addition and the integers with addition are isomorphic: the map $x \\mapsto 2x$ renames each integer as an even integer and turns sums into sums.\nThe integers and the rational numbers with addition are not isomorphic, although both are infinite: in $\\Q$ every element is twice another, and in $\\Z$ the element $1$ is not.\n",
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