Transition monoid

In mathematics and theoretical computer science, a semiautomaton is a deterministic finite automaton having inputs but no output. It consists of a set Q of states, a set Σ called the input alphabet, and a function T: Q × Σ → Q called the transition function.

Associated to any semiautomaton is a monoid called the characteristic monoid, input monoid, transition monoid or transition system of the semiautomaton, which acts on the set of states Q. This may be viewed either as an action of the free monoid of strings in the input alphabet Σ, or as the induced transformation semigroup of Q.

In older books like Clifford and Preston (1967) S-acts are called "operands".

In category theory, semiautomata essentially are functors.

A transformation semigroup or transformation monoid is a pair ${\displaystyle (M,Q)}$ consisting of a set Q (often called the "set of states") and a semigroup or monoid M of functions, or "transformations", mapping Q to itself. They are functions in the sense that every element m of M is a map ${\displaystyle m\colon Q\to Q}$. If s and t are two functions of the transformation semigroup, their semigroup product is defined as their function composition ${\displaystyle (st)(q)=(s\circ t)(q)=s(t(q))}$.

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