[Paper Review] Construction of schemoids from posets

Yasuhide Numata|arXiv (Cornell University)|Mar 2, 2016

semigroups and automata theory5 references3 citations

TL;DR

This paper constructs schemoid structures from posets by defining a small category from a poset and assigning morphisms based on order relations and rank functions. It proves that with a suitable labeling map, the resulting category forms a schemoid, and constructs its schemoid algebra as a quotient ring with relations derived from join operations and rank additivity, generalizing Bose–Mesner algebras for association schemes.

ABSTRACT

A schemoid is a generalization of association schemes from the point of view of small categories. In this article, we discuss schemoid structures for two kinds of small categories; the canonical small category defined by a poset, and another small category which arises a poset. We also discuss the schemoid algebra, that is an analogue of the Bose--Mesner algebra for an association scheme, for them.

Motivation & Objective

• To generalize association schemes via schemoids by extending the categorical framework to posets.
• To define a canonical small category from a poset that supports a schemoid structure.
• To construct the schemoid algebra as an analogue of the Bose–Mesner algebra for such schemoids.
• To establish conditions under which the morphism composition and labeling yield consistent cardinality conditions required for schemoid axioms.

Proposed method

• Define a small category $\tilde{P}$ from a poset $P$ where morphisms $f^d_{x,y}$ exist only if $y = x \vee d$ and $x \cap d = \emptyset$, with $\rho$ being a rank function.
• Assign a labeling map $\ell: \operatorname{Mor}(\tilde{P}) \to P$ by $\ell(f^d_{x,y}) = d$, which partitions morphisms into fibers.
• Verify the schemoid condition: for any $\sigma, \tau, \mu \in \ell^{-1}(P)$, the number of composable pairs $(f,g)$ with $f \circ g \in \mu$ is constant across all $h,k \in \mu$, using rank additivity $\rho(x \vee d) = \rho(x) + \rho(d)$.
• Define the schemoid algebra as the quotient ring $R_P = \mathbb{K}[X_x \mid x \in P]/I$, where $I$ is generated by relations $X_x X_y = X_{x \vee y}$ if $\rho(x \vee y) = \rho(x) + \rho(y)$, and $X_x X_y = 0$ otherwise.
• Prove that the schemoid algebra is isomorphic to $R_P$ by showing the multiplication rules match the composition cardinality conditions.
• Illustrate the construction on examples: Boolean lattices, subspaces of vector spaces over finite fields, and flats of matroids.

Experimental results

Research questions

• RQ1 Under what conditions on a poset $P$ can a schemoid structure be defined on its associated small category $\tilde{P}$?
• RQ2 How does the labeling map $\ell(f^d_{x,y}) = d$ ensure the schemoid condition on morphism composition?
• RQ3 What is the structure of the schemoid algebra associated with a poset-based schemoid, and how does it relate to known algebras like the Möbius algebra?
• RQ4 Can the schemoid algebra construction recover known algebras such as those in Maeno–Numata for matroids?
• RQ5 How do rank functions and join operations influence the algebraic structure of the schemoid algebra?

Key findings

• The triple $(\tilde{P}, P, \ell)$ forms a schemoid when $\ell(f^d_{x,y}) = d$ and the rank function satisfies $\rho(x \vee d) = \rho(x) + \rho(d)$ for all $x, d$ with $x \cap d = \emptyset$.
• The schemoid algebra is isomorphic to the quotient ring $R_P = \mathbb{K}[X_x \mid x \in P]/I$, where $I$ is generated by relations $X_x X_y = X_{x \vee y}$ if $\rho(x \vee y) = \rho(x) + \rho(y)$, and $X_x X_y = 0$ otherwise.
• For the $n$-th Boolean lattice $2^{[n]}$, the schemoid algebra corresponds to the algebra of the poset with $X_x X_y = X_{x \cup y}$ when $x \cap y = \emptyset$, and $X_x X_y = 0$ otherwise.
• For the poset of subspaces of a finite-dimensional vector space over a finite field, the schemoid algebra matches the structure of morphisms $f^W_{V,V \oplus W}$ with $V \cap W = 0$, and the algebra relations reflect the rank additivity of direct sums.
• For the poset of flats of a matroid satisfying the required conditions, the schemoid algebra is isomorphic to the Möbius algebra with $x_i^2 = 0$ for all generators, as defined in Maeno–Numata.
• The construction generalizes the Bose–Mesner algebra by replacing group-like symmetry with poset-based combinatorial structure via rank and join operations.