Invariants and Home Spaces in Transition Systems and Petri Nets

This lecture note focuses on comparing the notions of invariance and home spaces in Transition Systems and more particularly, in Petri Nets. We also describe how linear algebra relates to these basic notions in Computer Science, how it can be used for extracting invariant properties from a parallel system described by a Labeled Transition System in general and a Petri Net in particular. We endeavor to regroup a number of algebraic results dispersed throughout the Petri Nets literature with the addition of new results around the notions of semiflows and generating sets. \newline Several extensive examples are given to illustrate how the notion of invariants and home spaces can be methodically utilized through basic arithmetic and algebra to prove behavioral properties of a Petri Net. Some additional thoughts on invariants and home spaces will conclude this note.