Semiautomaton as a unary algebra

Example

A semiautomaton is a triple $(Q, \Sigma, \delta)$ with a set of states $Q$ and a set of input letters $\Sigma$. The map $\delta \colon Q \times \Sigma \to Q$ is the transition function. For each letter $a \in \Sigma$ put $\delta_a(q) = \delta(q, a)$. Then each $\delta_a$ maps $Q$ to $Q$, so $(Q, (\delta_a)_{a \in \Sigma})$ is a unary algebra with index set $\Sigma$. Conversely, 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.

Concretely, 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 $\mathrm{odd}$. Reading a binary word moves the state according to the parity of the number of letters $1$ read so far.

Under this translation, a homomorphism of unary algebras is a map of states that commutes with every transition. A subalgebra is a set of states closed under all transitions. A congruence is an equivalence of states compatible with all transitions. A deterministic finite automaton is a finite semiautomaton with a start state and a set of accepting states added.