Abstract A reaction system consists of a background set of entities and a set of reactions. Reactions are specified by three sets of entities: reactants, inhibitors, and products. A reaction is enabled by a state (a subset of entities), if all its reactants are present in that state and none of its inhibitors. The result of a set of reactions on a given state is a new state that consists of the products of the reactions that were enabled at the original state. In this paper, we further investigate enabling equivalence. This relation equates two sets of reactions for which the states that enable all their reactions simultaneously, are the same and, moreover, their results on those states are the same. From the point of view of enabling equivalence, sets of reactions act as if they were a single (combined) reaction. We show how combined reactions characterize enabling equivalence classes. Furthermore, enabling equivalence induces a partial order in the form of a cover relation on its equivalence classes. The resulting partially ordered set turns out to be a lattice and we demonstrate how this lattice relates to the enabling cover relation introduced earlier for single reactions.

Reaction systems are a computational model originally introduced to formalize the interactions between biochemical reactions that are the basis of the functioning of the living cell. Subsets of reactions of a reaction system define result functions which leads to a concept of functional equivalence. This equivalence in turn induces a functional cover relation on the reactions of a reaction system which captures redundancies in the system. In this paper, the functional cover relation is transferred to functional equivalence classes of sets of reactions. We introduce so-called atoms as building blocks of reactions. Atoms provide a characterization of functional equivalence classes of sets of reactions and are used to prove that the functional cover relation on functional equivalence classes of sets of reactions is a lattice.

For a family of sets we consider elements that belong to the same sets within the family as companions. The global dynamics of a reactions system (as introduced by Ehrenfeucht and Rozenberg) can be represented by a directed graph, called a transition graph, which is uniquely determined by a one-out subgraph, called the 0-context graph. We consider the companion classes of the outsets of a transition graph and introduce a directed multigraph, called an essential motion, whose vertices are such companion classes. We show that all one-out graphs obtained from an essential motion represent 0-context graphs of reactions systems with isomorphic transition graphs. All such 0-context graphs are obtained from one another by swapping the outgoing edges of companion vertices.

We investigate regular languages in the context of the forbidding-enforcing systems introduced by Ehrenfeucht and Rozenberg in the variant where one fe-system defines a single language. In general, these systems may have infinite sets of rules, allowing one to define arbitrary languages. On the oth er hand when restricted to finite sets, one obtains a strict subclass of the regular languages, between the strictly locally testable and locally testable languages. We further investigate classes of enforcing systems that characterize the regular languages. These systems have infinite sets of enforcers, but can be defined using regular languages (finite state automata).