We have arbitrary graphs Gx = (H, FI) and G z = (V, Fz), where H = {hi, I E I} and V = {vj, j E J}, while F1 and F z ate functions mapping H and V respectively into themselves. The graph G = (Q, F), which is the product GI • G2, is formed as follows [1]:~.where q E Q, h E H, and v ~ V, i . e . , Q = {qa, a E A ffi I x J}. We can extend the multiplication operation for graphs to finite automata. If Q and Z are, respectively, the sets of internal states and inputs [2], then an automaton can be given by a graph (called a graphoid) with weighted rays G = [Q, F(z E Z)]. Each ray of the graphoid denotes a transition from a state q a to ctB (qa, qB E Q) and is given a letre~ z ~ Z so that Fqa consists of aU pairs of the form c~(z), where z carries q a into q~. Let q0 E Q be the initial state of an automaton. Then the graphoid ~ completely and uniquely define some finite automaton, which we also denote by G.Consider two arbitrary automata Gz = [H, Ft(x E X)] and Gs = IV, F2(y E Y)]. Applying the multiplication operation to them, we obtain the automaton G = [Q, F, (z E Z)] = Gz x ~, which we call the product of the automata G1 and Fq=FzhxF~v, 0.9.) and q E Q, h E H, and v E V. If ho E H and vo E V are the initial states of GI and Gz, then the element (h o, vo) = qo E E Q is the initial state of G.Note that every automaton can be represented as the union of autonomous automata [3], L e . , ^ o, ffi H, F,. ffi Ua, , rex a, ---Ua,,,.Then the resulting automaton G ffi GI X G2 can also be written in the form a (u a,,/-.
L~x ! \~" /Ix, ttIEXxF zEZ It is not difficult to see that the operation of multiplying graphs is associative and commutes up to isomorphism.Since the multiplication operation is applicable to drbitrary graphs, the set r of graphs obtained by multiplication is a sernigroup with a unit, whose, role is played by the graph G' = (W, F'), where Q' ={q} and F'q ={q}. Any graph G E E r can be expanded in a trivial way into the product G = G X G ~ ffi G' x G up to isomorphism. The subset r ! c r of graphs which can be expandecl nontrivially into the product of graphs in r is a subsemigroup of the semigroup r. In this paper we solve the problem of deciding whether an arbitrary graph G E r belongs to the subsemigroup r I c r and we describe a method for finding possible expansions of a given graph G E r I into a product of graphs in r.We use out results to extract, from a sernigroup r, subsemigroups rl c r of automata, which can each be represented as the product of automata in r.
The Expansion of a Graph into a Product of GraphsBefore we state a theorem ona necessary and sufficient condition for the expansion of a graph into the product of two graphs, we will discuss certain concepts which we shall need later.