a function called the transition function. Semiautomata arise from deterministic automata by ignoring the initial state and the set of accept states. Given a semiautomaton, let Ta: X → X, for a ∈ Σ, denote the transformation of X defined by Ta(x) = T(a,x). Then the semigroup of transformations of X generated by {Ta : a ∈ Σ} is called the characteristic semigroup or transition system of (Σ,X,T). This semigroup is a monoid, so this monoid is called the characteristic or transition monoid. It is also sometimes viewed as a Σ∗-act on X, where Σ∗ is the free monoid of strings generated by the alphabet Σ, and the action of strings extends the action of Σ via the property Krohn–Rhodes theory Krohn–Rhodes theory, sometimes also called algebraic automata theory, gives powerful decomposition results for finite transformation semigroups by cascading simpler components. Notes References A. H. Clifford and G. B. Preston (1961), The Algebraic Theory of Semigroups, volume 1. American Mathematical Society, . A. H. Clifford and G. B. Preston (1967), The Algebraic Theory of Semigroups, volume 2. American Mathematical Society, . Mati Kilp, Ulrich Knauer, Alexander V. Mikhalev (2000), Monoids, Acts and Categories: with Applications to Wreath Products and Graphs, Expositions in Mathematics 29, Walter de Gruyter, Berlin, . Rudolf Lidl and Günter Pilz, Applied Abstract Algebra (1998), Springer, Semigroup theory Theoretical computer science