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  "body": "\n## Example\n\nA *Jónsson–Tarski algebra* is an [[algebra|QJGPT2W0FT]] $(A, \\cdot, L, R)$ with a binary operation $\\cdot$ and two [[unary operations|Q2W26ZY10B]] $L$ and $R$ such that, for all $x, y \\in A$:\n\n1. $L(x \\cdot y) = x$;\n2. $R(x \\cdot y) = y$;\n3. $L(x) \\cdot R(x) = x$.\n\nThese laws say exactly that $(x, y) \\mapsto x \\cdot y$ is a [[bijection|0GG9TVW90M]] from $A \\times A$ to $A$, with inverse $x \\mapsto (L(x), R(x))$.\nSince the laws are identities, Jónsson–Tarski algebras form a [[variety|AJNZJD8442]].\n\nA concrete instance is the set $A$ of infinite binary sequences $s = (s_0, s_1, s_2, \\dots)$.\nLet $s \\cdot t = (s_0, t_0, s_1, t_1, \\dots)$ interleave the two sequences.\nLet $L(s) = (s_0, s_2, s_4, \\dots)$ and $R(s) = (s_1, s_3, s_5, \\dots)$.\nThen $L(s \\cdot t) = s$ and $R(s \\cdot t) = t$ because the even places of $s \\cdot t$ hold $s$ and the odd places hold $t$.\nAlso $L(s) \\cdot R(s) = s$ because interleaving the even and odd places rebuilds $s$.\n\n## Remarks\n\nA Jónsson–Tarski algebra with more than one element is infinite.\nIndeed, if $A$ is finite with $n$ elements, a bijection $A \\times A \\to A$ forces $n^2 = n$, so $n \\le 1$.\nJónsson and Tarski showed that in this variety the [[free algebra|GTS4DRRMZR]] on one generator is [[isomorphic|PZMGQMDN25]] to the free algebra on $n$ generators for every finite $n \\ge 1$.\nSo the number of free generators is not an invariant here, unlike for vector spaces.\n\n## Explanations\n\n### L1 human\n\nA Jónsson–Tarski algebra is a set with a perfect way to pack two elements into one and unpack them again.\nPacking $x$ and $y$ gives $x \\cdot y$, and the maps $L$ and $R$ recover the left and right parts.\nEvery element arises from exactly one pair in this way.\nInfinite binary sequences give an example: shuffle two sequences together like two decks of cards, and unshuffle to get them back.\nNo finite set with more than one element can do this, since it has more pairs than elements.\n",
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