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  "body": "\n## Summary\n\nThe congruence generated by a relation $R$ on an [[algebra|QJGPT2W0FT]] is the smallest [[congruence|DCSXXT4SER]] that contains $R$.\nIt exists because any intersection of congruences is a congruence.\nIt is the tool for forcing given pairs of elements to become equal while keeping the operations meaningful, as in presentations by generators and relations.\n\n## Explanations\n\n### L1 human\n\nSuppose you want some pairs of elements of an algebra to count as equal.\nOnce $a$ counts as equal to $b$, compatibility forces many other pairs to be equal too, such as $a + c$ and $b + c$.\nThe congruence generated by your pairs collects exactly what is forced and nothing more.\nIn the group of integers under addition, asking for $0$ to equal $6$ forces exactly congruence modulo $6$.\n",
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