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  "body": "\n## Summary\n\nA unary algebra is a set with a family of [[unary operations|Q2W26ZY10B]], maps from the set to itself, and no operations of other arities.\nIt is the simplest kind of [[algebraic structure|QJGPT2W0FT]], and yet it already contains the theory of functional graphs and of automata.\nA unary algebra with operations indexed by a set $I$ is the same as an action of the free monoid on $I$.\nA unary algebra with one operation is called a mono-unary algebra.\n\n## Explanations\n\n### L1 human\n\nTake a set and some ways of moving each element to another element.\nThat is a unary algebra.\nWith one operation $f$, draw an arrow from each $x$ to $f(x)$, so that every point has exactly one outgoing arrow.\nFor a finite set, the picture is made of cycles with trees hanging off them.\nFor example, the integers with the successor map $n \\mapsto n + 1$ form a unary algebra whose picture is one infinite line.\nA [[semiautomaton|RTMYXRDGWR]] is a unary algebra with one operation for each input letter.\n",
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