Title: On Invariants and Substution Systems

Invariants of algebraic nets are structures that describe invariance properties of the net. There are essentially two ways of defining invariants for algebraic nets. In the first method one unfolds the high-level net into its representation as a low-level net and does the calculations there. The problem is thus reduced to finding invariants of this low-level net. The second method views algebraic nets as a formal system for themselves, where the semantics is defined in terms of substitutions. In this approach invariants I are derived as solutions to the matrix equation M . I = 0 where M is the incidence matrix of the net. The product operation is a complex operator that is a combination of substitution and the operations of a group. It is the aim of this work to discuss the latter method in the categorical framework of Substitution Systems as proposed by Goguen. The emphasis is on the definition of a suitable substitution system for the terms labeling the net. This substitution system is constructed as a Kleisli category. The incidence matrix defines a set of arrows, one arrow for each transition, in this Kleisli category. An invariant is then a substitution that is the solution to an equation on the arrows of the Kleisli category.