Causal Semantics of Algebraic Petri Nets distinguishing Concurrency and Synchronicity

In this paper, we show how to obtain causal semantics distinguishing "earlier than" and "not later than" causality between events from algebraic semantics of Petri nets. Janicki and Koutny introduced so called stratified order structures (so-structures) to describe such causal semantics. To obtain algebraic semantics, we redefine our own algebraic approach generating rewrite terms via partial operations of synchronous composition, concurrent composition and sequential composition. These terms are used to produce so-structures which define causal behavior consistent with the (operational) step semantics. For concrete Petri net classes with causal semantics derived from processes minimal so-structures obtained from rewrite terms coincide with minimal so-structures given by processes. This is demonstrated for elementary nets with inhibitor arcs.