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  "body": "\n## Summary\n\nA homomorphism is a map between two [[structures|983GZBQYT4]] of the same [[signature|WDVVEYD6AT]] that preserves the operations, the constants and the relations.\n[[Group homomorphisms|0JYTG5K9Y4]], [[ring homomorphisms|0VTH8KPCG2]] and [[linear maps|0NRYHJPQG2]] are all homomorphisms for suitable signatures.\nHomomorphisms are the maps used to compare structures: [[isomorphisms|PZMGQMDN25]], [[embeddings|SCRB74SWQN]] and [[quotient maps|N5KE7FH6CX]] are special homomorphisms.\n\n## Explanations\n\n### L1 human\n\nA homomorphism translates one structure into another without breaking its rules.\nIf $h$ is a homomorphism for addition, then computing $a + b$ and then applying $h$ gives the same as applying $h$ to $a$ and $b$ and then adding.\n[[Reduction of integers modulo a fixed number|BV08DGBD7C]] is the standard example: the remainder of a sum is determined by the remainders of the summands.\nThe map $x \\mapsto x + 1$ on the integers is [[not a homomorphism|3A3JJPJHQR]] for addition, since it does not send $0$ to $0$.\n",
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