Conway and iteration hemirings

TL;DR

This paper introduces Conway and iteration hemirings, explores their relationships, and applies hemimodules to analyze quantitative behaviors in weighted transition systems, including discounted and average computations.

Contribution

It defines and studies Conway and iteration hemirings, their relationships, free constructions, and their application to quantitative analysis of weighted transition systems.

Findings

Established relationships between Conway hemirings and partial Conway semirings.

Developed free constructions for these algebraic structures.

Applied hemimodules to analyze weighted transition systems' behaviors.

Abstract

Conway hemirings are Conway semirings without a multiplicative unit. We also define iteration hemirings as Conway hemirings satisfying certain identities associated with the finite groups. Iteration hemirings are iteration semirings without a multiplicative unit. We provide an analysis of the relationship between Conway hemirings and (partial) Conway semirings and describe several free constructions. In the second part of the paper we define and study hemimodules of Conway and iteration hemirings, and show their applicability in the analysis of quantitative aspects of the infinitary behavior of weighted transition systems. These include discounted and average computations of weights.