Operad of posets 101: The Wixárika posets
José Antonio Arciniega-Nevárez, Marko Berghoff, Eric Rubiel Dolores-Cuenca
TL;DR
The paper introduces Wixárika posets, a targeted subfamily of series-parallel posets generated from a single element by the operations $D$ and $*$, and situates them within the operadic framework. It defines and studies order series $\mathfrak{Z}(P)$ as algebras over the Wixárika operad $W$, establishing a homomorphism between Wixárika posets and their order-series via $\mathfrak{Z}$ and detailing how the operad governs their algebraic and topological structure. An explicit evaluation method computes $\mathfrak{Z}(P)$ from tree representations, enabling extraction of $\Omega^{\circ}(m)$, while a representability analysis uses topological invariants (e.g., Betti numbers) to solve the inverse problem and identify all posets representing a given $f(x)$. The work highlights how operad theory links combinatorics, topology, and algebra to provide concrete computational tools for the order-series of series-parallel posets and outlines directions for broader operadic exploration and applications.
Abstract
We study sets whose combinatorics are related to the combinatorics of posets. The language of operads provides us with tools to better understand the combinatorics of these objects. In this note we describe a non-trivial example of a suboperad, called the Wixárika posets, together with its associated algebras. This example is rich enough to showcase the particularities of the field, without delving into technicalities.