Algebraic Theory of Machines. I. Prime Decomposition Theorem for Finite Semigroups and Machines

Introduction. In the following all semigroups are of finite order. One semigroup Si is said to divide another semigroup S2, written Si|S2, if Si is a homomorphic image of a subsemigroup of S2. The semidirect product of S2 by Si, with connecting homomorphism Y, is written S2 Xy Si. See Definition 1.6. A semigroup S is called irreducible if for all finite semigroups S2 and Si and all connecting homomorphisms Y, S\(S2Xy Si) implies S|S2 or S|Si. It is shown that S is irreducible if and only if either: (i) S is a nontrivial simple group, in which case S is called a prime; or (ii) S is one of the four divisors of a certain three element semigroup U3 (see Definition 2.1) in which case S is called a unit. We remark that an anti-isomorphism of a unit need not be a unit. Thus the theory is not symmetric. The explanation is that semidirect product can be written from the left or from the right. Let be a collection of finite semigroups. We define K(Sf) as the closure of Sunder the operations of division and semidirect product. See Definition 3.2. Then it is proved that SG K{Sf U {U3\) if and only if PRIMES (S) C PRIMES (SS). Here PRIMES (S) = {P\ P is a nontrivial simple group and P divides S\ and PRIMES (SS) =(J {PRIMES (S)\S G Sf ). In particular, Se#(PRIMES (S) U {U3\). A counterexample to the conjecture that SG 7/f(IRR(S)) justifies the distinction between primes and units as well as the inclusion of U3 in the above formulas. A novel feature of this paper is the use of functions on free semigroups, i.e. machines, to prove facts about finite semigroups. These above results are obtained as an immediate corollary of a more general theorem (proved here) which finds application as the basis for a prime decomposition theorem for finite state sequential machines. Further, by applying this theorem together with the powerful solvability criteria of Feit and Thompson and of Burnside, we find that Corollary 4.1 answers in important cases the question "What machines can be constructed by series-parallel from counters, delays and units?" See §4. A heuristic discussion of this paper occurs in [6].