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  "body": "\n## Summary\n\nA universal class is a class of structures axiomatized by universal sentences, those of the form \"for all $x_1, \\ldots, x_n$, a quantifier-free condition holds\".\nUniversal sentences are preserved by [[substructures|676H7EGTKJ]], so every universal class is closed under substructures.\nBy the [[Łoś–Tarski preservation theorem|9RN0KYXNT1]], an [[elementary class|MEC3ZMC4F1]] is universal exactly when it has this closure property.\nVarieties, quasivarieties and universal Horn classes are universal classes.\n\n## Explanations\n\n### L1 human\n\nA universal axiom only checks finite patterns of elements, one tuple at a time.\nIf a structure passes every such check, then so does every part of it, since a part has fewer tuples to check.\nSimple graphs form a universal class: \"no vertex is adjacent to itself\" and \"adjacency is symmetric\" are universal.\nFields do not: the axiom \"every nonzero element has an inverse\" needs an \"exists\", and indeed the subring $\\Z$ of the field $\\Q$ is not a field.\n",
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