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  "body": "\n## Definition\n\nLet $n \\ge 2$ and let $P_n = \\set{0, 1, \\dots, n - 1}$.\nThe *$n$-element Post algebra* is $\\mathbf{P}_n = (P_n, \\vee, \\wedge, ', 0, n - 1)$.\nHere $x \\vee y = \\max(x, y)$, $x \\wedge y = \\min(x, y)$, and $x' = x + 1$ for $x \u003c n - 1$ while $(n - 1)' = 0$.\nA *Post algebra of order $n$* is an algebra in the [[variety|AJNZJD8442]] generated by $\\mathbf{P}_n$.\n\n## Remarks\n\nExample: $\\mathbf{P}_n^2$ is a Post algebra of order $n$, since a variety is closed under direct products.\nNon-example: the three-element chain $(\\set{0, 1, 2}, \\max, \\min, ', 0, 2)$ with $x' = 2 - x$ is not a Post algebra of order $3$.\nIn $\\mathbf{P}_3$ the identity $x''' \\approx x$ holds, but here $0''' = 2 \\ne 0$.\nThe algebra $\\mathbf{P}_n$ is [[primal|XD01Y7B6XW]], because $\\max$ and the cyclic negation already generate every operation on $P_n$, a result of Post.\nFor $n = 2$, $\\mathbf{P}_2$ is the two-element Boolean algebra, so Post algebras of order $2$ are Boolean algebras.\nOther texts present Post algebras through lattice axioms, as bounded distributive lattices with a chain of constants $e_0 \u003c e_1 \u003c \\cdots \u003c e_{n - 1}$.\nPost algebras give algebraic semantics for Post's [[many-valued logic|DWYCKAKXZD]].\n",
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