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  "body": "\n## Definition\n\nLet $A$ be a [[set|0SJ3RX9TXA]] and let $I$ be an index set.\nA *unary algebra* is a pair $(A, (f_i)_{i \\in I})$, where each $f_i \\colon A \\to A$ is a [[unary operation|Q2W26ZY10B]] on $A$.\nIt is *mono-unary* when $I$ has exactly one element.\nA *homomorphism* of unary algebras $(A, (f_i))$ and $(B, (g_i))$ with the same index set is a [[function|0EPAT51DXQ]] $h \\colon A \\to B$ with $h(f_i(a)) = g_i(h(a))$ for all $i \\in I$ and $a \\in A$.\n\n## Remarks\n\nExample: $(\\Z, s)$ with $s(n) = n + 1$ is a mono-unary algebra, since $s$ maps $\\Z$ to $\\Z$.\nThe map $h \\colon \\Z \\to \\Z/2\\Z$, $h(n) = n \\bmod 2$, is a homomorphism to $(\\Z/2\\Z, t)$ with $t(x) = x + 1$, because $h(n + 1) = h(n) + 1$.\n\nNon-example: $(\\N, p)$ with the predecessor $p(n) = n - 1$ is not a unary algebra, since $p(0) = -1 \\notin \\N$; the predecessor is only a partial operation on $\\N$.\n\nA unary algebra $(A, (f_i)_{i \\in I})$ is the same as an action of the free monoid $I^*$ on $A$: a word $i_1 i_2 \\cdots i_k$ acts by $f_{i_k} \\circ \\cdots \\circ f_{i_1}$ in one common convention.\n",
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