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  "body": "\n## Example\n\nA *semiautomaton* is a triple $(Q, \\Sigma, \\delta)$ with a set of states $Q$ and a set of input letters $\\Sigma$.\nThe map $\\delta \\colon Q \\times \\Sigma \\to Q$ is the transition function.\nFor each letter $a \\in \\Sigma$ put $\\delta_a(q) = \\delta(q, a)$.\nThen each $\\delta_a$ maps $Q$ to $Q$, so $(Q, (\\delta_a)_{a \\in \\Sigma})$ is a [[unary algebra|XD10P9CJMD]] with index set $\\Sigma$.\nConversely, a unary algebra $(Q, (f_a)_{a \\in \\Sigma})$ gives the semiautomaton with $\\delta(q, a) = f_a(q)$, so the two notions are the same data.\n\nConcretely, let $Q = \\set{\\mathrm{even}, \\mathrm{odd}}$ and $\\Sigma = \\set{0, 1}$, with $\\delta_0$ the identity of $Q$ and $\\delta_1$ the map that swaps $\\mathrm{even}$ and $\\mathrm{odd}$.\nReading a binary word moves the state according to the parity of the number of letters $1$ read so far.\n\nUnder this translation, a homomorphism of unary algebras is a map of states that commutes with every transition.\nA subalgebra is a set of states closed under all transitions.\nA congruence is an equivalence of states compatible with all transitions.\nA deterministic finite automaton is a finite semiautomaton with a start state and a set of accepting states added.\n\n## Explanations\n\n### L1 human\n\nA semiautomaton is a machine that changes its state each time it reads a letter.\nFor each letter there is a rule that sends every state to a new state.\nSo each letter is a map from the set of states to itself, and the machine is a set with a family of such maps.\nThat is exactly a unary algebra, so the tools of universal algebra, such as quotients and homomorphisms, apply directly to machines.\n",
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