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  "body": "\n## Summary\n\nA cylindric algebra is a [[Boolean algebra|0XADJKM699]] with extra operations that model existential quantifiers and constants that model equality of variables.\nAlfred Tarski and his collaborators introduced them to do for [[predicate logic|086QKPF0A6]] with equality what Boolean algebras do for propositional logic.\nThe cylindrification $c_\\kappa$ plays the role of \"there exists $x_\\kappa$\", and the diagonal element $d_{\\kappa\\lambda}$ plays the role of the formula $x_\\kappa = x_\\lambda$.\n\n## Explanations\n\n### L1 human\n\nThink of the set of points of the plane that satisfy a formula $\\varphi(x, y)$.\nThe formula \"there exists $x$ such that $\\varphi(x, y)$\" describes the set swept out when that region is moved along the $x$-direction: a cylinder over the shadow of the region.\nThat is why the operation is called cylindrification.\nThe diagonal line $\\set{(x, y) : x = y}$ is the set of the formula $x = y$.\nA cylindric algebra keeps only the rules that these operations obey, so it can be studied with algebra.\n",
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