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  "body": "\n## Statement\n\nLet $\\sigma$ be a first-order signature and let $T$ be a first-order [[theory|92SWXST1ZM]] in $\\sigma$.\nThen the following are equivalent.\n\n1. The class of [[models|MVCH6N9SNW]] of $T$ is closed under [[substructures|676H7EGTKJ]]: if $B \\models T$ and $A$ is a substructure of $B$, then $A \\models T$.\n2. There is a set $U$ of universal sentences of $\\sigma$ with the same models as $T$; so the models of $T$ form a [[universal class|ZSTEZ3PRT9]].\n\n## Remarks\n\nThe direction from 2 to 1 holds because universal sentences pass from a structure to its substructures.\nFor the other direction, let $U$ be the set of universal sentences that follow from $T$, and let $A \\models U$.\nExpand the signature by a constant for each element of $A$, and let $D$ be the set of quantifier-free sentences true in $A$, its diagram.\nIf $T \\cup D$ had no model, then by the [[compactness theorem|1NR3HAWG9D]] $T$ would prove the negation of a finite conjunction from $D$.\nReplacing the new constants by variables gives a universal sentence in $U$ that fails in $A$.\nSo $T \\cup D$ has a model $B$, into which $A$ embeds, and condition 1 gives $A \\models T$.\n\nA version for formulas holds too: a formula is preserved by substructures between models of $T$ exactly when it is equivalent modulo $T$ to a universal formula.\nDually, the class of models of $T$ is closed under extensions exactly when $T$ is equivalent to a set of existential sentences.\n\nExample: the theory of groups in the signature $(\\cdot, {}^{-1}, e)$ is closed under substructures, and the group axioms are indeed universal.\nIn the signature $(\\cdot)$ alone, a substructure of a group is only a subsemigroup, such as $\\N_{\u003e0}$ in $(\\Z, +)$, and the axioms of groups in this signature need existential quantifiers.\n\nThe theorem uses compactness, and it fails when only finite structures are considered.\n\n## Explanations\n\n### L1 human\n\nA universal axiom says that something holds for all elements, without asking for any element to exist.\nSuch axioms survive passing to a part of a structure.\nThe theorem says the converse: if a first-order axiom system always survives passing to parts, then it can be rewritten with universal axioms only.\nSo a semantic property, closure under substructures, matches a syntactic one, the shape of the axioms.\n\n### L2 human\n\nThe Łoś–Tarski theorem is the first of the preservation theorems of model theory.\nOthers in the same spirit are the Chang–Łoś–Suszko theorem, for unions of chains and $\\forall\\exists$ sentences, and Lyndon's theorem, for homomorphic images and positive sentences.\nIn universal algebra it explains why an elementary class of algebras closed under subalgebras has a universal axiomatization.\nThis is a first step towards the descriptions of quasivarieties and varieties.\nThe proof method, realizing the diagram of a structure in a model of $T$, is the method of diagrams used throughout model theory.\n",
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