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  "body": "\n## Summary\n\nA universal Horn class is the class of all structures of a signature that satisfy a given set of universal Horn sentences.\nSuch a sentence is a universally quantified [[Horn clause|QR92EZDTKF]] built from first-order atomic formulas.\nUniversal Horn classes are [[universal classes|ZSTEZ3PRT9]], and in addition they are closed under direct products of nonempty families.\nIn an algebraic signature, the classes axiomatized by universal Horn sentences with a positive literal are exactly the [[quasivarieties|WRXEKT4NQS]].\n\n## Explanations\n\n### L1 human\n\nA universal Horn sentence says: for all elements, if these basic facts hold, then that basic fact holds too, or else that these facts never hold together.\nPartial orders form a universal Horn class: reflexivity, antisymmetry and transitivity all have this shape.\nTotal orders do not, because \"$x \\le y$ or $y \\le x$\" offers a choice between two facts.\nIndeed the product of two 2-element chains is a square, in which two corners are incomparable.\n",
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