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  "body": "\n## Definition\n\nLet $\\sigma$ be a first-order [[signature|WDVVEYD6AT]].\nA *universal sentence* of $\\sigma$ is a [[sentence|DKFGQTRYZE]] of the form $\\forall x_1 \\cdots \\forall x_n \\, \\varphi$, with $n \\ge 0$ [[universal quantifiers|0GHS47PDAQ]].\nHere $\\varphi$ is a quantifier-free formula whose free variables are among $x_1, \\ldots, x_n$.\nLet $K$ be a class of [[$\\sigma$-structures|983GZBQYT4]].\nThen $K$ is a *universal class* if there is a set $\\Sigma$ of universal sentences of $\\sigma$ whose models are exactly the members of $K$.\n\n## Remarks\n\nExample: simple graphs, in the signature with one binary relation $E$, form a universal class with the axioms $\\forall x \\, \\neg E(x, x)$ and $\\forall x \\forall y \\, (E(x, y) \\rightarrow E(y, x))$.\n\nNon-example: fields, in the signature $(+, -, \\cdot, 0, 1)$, do not form a universal class.\nA universal sentence true in a structure is true in each [[substructure|676H7EGTKJ]], because it only speaks about tuples of elements and the operations on them.\nThe ring $\\Z$ is a substructure of the field $\\Q$ in this signature, and it is not a field.\n\nEvery universal class is an [[elementary class|MEC3ZMC4F1]].\nThe [[Łoś–Tarski preservation theorem|9RN0KYXNT1]] gives the converse characterization: an elementary class is universal exactly when it is closed under substructures.\n",
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