Summary: A primitive product is a composition of a finite sequence of finite automata such that feedback is limited to no further than the previous factor. Furthermore, the input to each factor depends only on the global input to the system and the states of at most three factors (including the factor itself). Conversely, the state of a factor may directly influence only at most three factors (including the factor itself). Additional conditions guarantee a strong planarity property known as outerplanarity. Any primitive product of automata can be realized in an outerplanar layout. This is desirable from the engineering point of view of simple circuit wiring as a circuit whose components and wires comprise the nodes and edges of an outerplanar graph may be realized on a two-dimensional surface, and moreover, new wires can be run from a point outside the circuit to any or all nodes of the circuit without crossing each other or any existing wires.
We constructively show that if \(A4\) is a finite automaton satisfying Letichevsky’s criterion, then any finite automaton can be homomorphically represented by (i.e. is a homomorphic image of a subautomaton of, or equivalently, is a letter-to-letter [length-preserving] divisor of) a primitive product of copies of \(A\). A class \(K\) of finite automata is homomorphically complete under a given product \(\pi\), by definition, if every finite automaton can be homomorphically represented as a \(\pi\)-product of automata from \(K\). By Letichevsky’s characterization of homomorphically complete classes under the general product (unrestricted finite composition), our results imply that a class of finite automata is homomorphically complete under the general product if and only if it is homomorphically complete under the primitive product.