Petri Nets and its Polynomials

TL;DR

This paper introduces a polynomial representation for finite Petri nets, establishing a correspondence between polynomial multiplication and Petri net composition, and explores the topological structure of Petri nets.

Contribution

It presents a novel polynomial encoding of Petri nets, an inverse construction, and links algebraic operations to Petri net composition, enriching the mathematical framework of Petri net theory.

Findings

Polynomial multiplication corresponds to Petri net product

Set of Petri nets forms a Zariski topology

Inverse construction for Petri nets from polynomials

Abstract

For every finite Petri net, we construct a commutative polynomial in two variables and with coefficients from the semiring of natural numbers. We also present an inverse construction and show that multiplication of polynomials correspondence to the product of the corresponding Petri nets in the category of Petri nets with Winskel's morphisms. We endow the set of all Petri nets with Zariski topology.