Operad of posets 101: The Wixárika posets

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.

💡 Research Summary

The paper “Operad of posets 101: The Wixárika posets” presents a pedagogical yet substantive study of a particular sub‑operad of finite posets, called Wixárika posets, and explores the algebras over this operad. After a brief motivation, the authors recall the standard construction of the operad Poset whose n‑ary elements are finite posets with n elements, and whose composition replaces a chosen vertex of a poset by the Hasse diagram of another. This sets the stage for a more concrete operadic framework built from two elementary operations: the binary series composition ∗ (direct sum) and a unary “itsari” operation D.

The itsari operation D adds a new global minimum x₀, a new global maximum x₁, and a single intermediate element y with x₀ < y < x₁, effectively “weaving” a handle onto the poset. Starting from the one‑element poset, any finite Wixárika poset can be obtained by finitely many applications of ∗ and D. Consequently, the collection W generated by these two operations forms an operad whose elements can be visualized as rooted binary trees whose internal nodes are labelled by ∗ and whose unary nodes (the “bird nests”) are labelled by D.

To connect combinatorial data with algebraic structures, the authors introduce order‑preserving and strictly order‑preserving maps from a poset P to the chain ⟨n⟩, denoted Ω∘(P,n) and Ω(P,n) respectively. The “order series” of P is defined as the formal power series
 Z(P,x)=∑_{n≥1} Ω∘(P,n) xⁿ.
Standard operations on power series—Cauchy product, direct product (defined as a Cauchy product with a factor (1−x) ), and Hadamard product—are shown to correspond precisely to the operadic operations ∗ and ⊔ (disjoint union) on posets. In particular, the identities
 Z(P∗Q)=Z(P)·(1−x)·Z(Q) and Z(P⊔Q)=Z(P)⊙Z(Q)
hold for all finite posets. Moreover, the itsari operation is linear on order series: D(a Z(P)+b Z(Q))=a D(Z(P))+b D(Z(Q)).

An algebra over an operad O is defined as an operad morphism O→End(A), where End(A)(n)=Map(Aⁿ,A). The paper defines a W‑algebra as a set equipped with concrete realizations of the abstract operations ∗ and D. Two natural W‑algebras are considered: the set of Wixárika posets themselves (with composition given by the operadic substitution) and the set of order series (with the series operations described above). The map Z: W‑posets → W‑order‑series is proved to be an operad morphism, i.e., it respects both ∗ and ⊔ as well as the itsari operation.

The authors then work out an explicit example: starting from the one‑element chain ⟨1⟩, they apply a sequence of D and ∗ operations to obtain a non‑trivial Wixárika poset. Using the established identities, they compute its order series step by step, illustrating how the algebraic machinery simplifies what would otherwise be a cumbersome enumeration of order‑preserving maps. The example also demonstrates the linearity of D on series and the way series‑parallel decompositions translate into linear combinations of chain series.

Finally, the paper discusses how topological information encoded in the Hasse diagram (such as height, width, and connectivity) can be transferred to the algebraic level via the operadic framework. Although the authors do not present new theorems, the exposition showcases how operads provide a unifying language for combinatorial constructions, generating functions, and algebraic representations. The work serves as a gentle introduction to operadic methods for combinatorialists and suggests future directions, including the study of more complex operads, non‑binary operations, and connections with Hopf algebras of posets.