Krohn–Rhodes theorem

In mathematics and computer science, the Krohn–Rhodes theory (or algebraic automata theory) is an approach to the study of finite semigroups and automata that seeks to decompose them in terms of elementary components. These components correspond to finite aperiodic semigroups and finite simple groups that are combined together in a feedback-free manner (called a "wreath product" or "cascade").

Krohn and Rhodes found a general decomposition for finite automata. In doing their research, though, the authors discovered and proved an unexpected major result in finite semigroup theory, revealing a deep connection between finite automata and semigroups.

A semigroup S that is a homomorphic image of a subsemigroup of T is said to be a divisor of T.

The Krohn–Rhodes theorem for finite semigroups states that every finite semigroup S is a divisor of a finite alternating wreath product of finite simple groups, each a divisor of S, and finite aperiodic semigroups (which contain no nontrivial subgroups).

In the automata formulation, the Krohn–Rhodes theorem for finite automata states that given a finite automaton A with states Q and input set I, output alphabet U, then one can expand the states to Q' such that the new automaton A' embeds into a cascade of "simple", irreducible automata: In particular, A is emulated by a feed-forward cascade of (1) automata whose transitions semigroups are finite simple groups and (2) automata which are banks of flip-flops running in parallel. The new automaton A' has the same input and output symbols as A. Here, both the states and inputs of the cascaded automata have a very special hierarchical coordinate form.

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