TOL Schemes and Control Sets

Suppose given two of the following: a set L1 of start words, a set L2 of target words, and a control set C of finite sequences of applications of a given finite set of homomorphisms (or finite substitutions) which map L1 into L2. Using notions from OL systems, the present paper investigates what can be said about the remaining set in case the given sets are regular. When the start and target sets are regular, the set of all control words turns out to be regular. (This is true even when the regularity assumption on the start set is removed.) When a regular target set L2 and a regular control set C are given, the set of all words map ped into L2 by C is regular. (This result remains true even when the regularity assumption on C is removed.) When a regular start set L and a regular control set C are given, the set C (L) is an ETOL language. In fact, this characterizes ETOL languages. Finally, it is shown that the set ∋(∑) of all possible homomorphisms (or the set C (∑) of all finite substitutions) from a given alphabet ∑ into itself cannot be a control set. In other words, neither of the semigroups ∋(∑) or C (∑) is finitely generated.