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  "body": "\n## Summary\n\nA natural duality represents each algebra in the [[quasivariety|WRXEKT4NQS]] generated by a finite algebra $M$.\nThe algebra becomes an algebra of continuous structure-preserving maps on a topological structure.\nThe dual of an algebra $A$ is the set of homomorphisms from $A$ to $M$.\nIt is viewed inside a power of a topological structure $\\underset{\\sim}{M}$ on the same set as $M$.\n[[Stone duality|B9T33D5MJK]] for Boolean algebras and [[Priestley duality|SA22SEC75K]] for bounded distributive lattices are the model cases.\nThe theory gives general conditions under which such a duality exists.\nIt also tells when the duality is a dual equivalence of categories, a [[Stone-type duality|VH12PN9V13]].\n\n## Explanations\n\n### L1 human\n\nStone showed that a Boolean algebra can be rebuilt from the set of its homomorphisms into the two-element Boolean algebra, once that set carries the right topology.\nNatural duality asks the same question for other finite \"building block\" algebras $M$.\nEach algebra $A$ built from copies of $M$ is turned into a space of homomorphisms $A \\to M$, with extra relations or operations remembered from $M$.\nThe goal is that $A$ can be recovered as the continuous structure-preserving maps from that space back to $M$.\n",
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