Coproducts and decomposable machines

摘要

The crucial discovery reported here is that the free monoid U* on the input set U does not yield a sufficiently rich set of inputs when algebraic structure is placed on the machine. For group machines, the appropriate structure is the coproduct U§ of an infinite sequence of copies of U. U§ reduces to a reasonable facsimile of U* in the Abelian case. A structure theorem for monoids of linear systems reveals the R monoid of Give'on and Zalcstein as appropriate only when no distinct powers of the statetransition matrix have the same action.