Generating Posets Beyond N

TL;DR

This paper introduces iposets, a new algebraic structure for posets with interfaces, enabling the generation of complex posets like series-parallel and interval orders, which are useful for modeling concurrent systems.

Contribution

The paper defines iposets with gluing composition, studies their algebraic properties, and demonstrates their ability to generate important classes of posets beyond N.

Findings

iposets can generate series-parallel posets

iposets can generate interval orders

not all posets can be generated by iposets

Abstract

We introduce iposets---posets with interfaces---equipped with a novel gluing composition along interfaces and the standard parallel composition. We study their basic algebraic properties as well as the hierarchy of gluing-parallel posets generated from singletons by finitary applications of the two compositions. We show that not only series-parallel posets, but also interval orders, which seem more interesting for modelling concurrent and distributed systems, can be generated, but not all posets. Generating posets is also important for constructing free algebras for concurrent semirings and Kleene algebras that allow compositional reasoning about such systems.