Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Operad Structure of Poset Matrices

Arnauld Mesinga Mwafise, Gi-Sang Cheon, Hong Joon Choi, Samuele Giraudo

Abstract

This paper examines operad structures derived from poset matrices by formulating a set of new construction rules for poset matrices. In this direction, eleven different partial composition operations will be introduced as the basis for the construction of poset matrices of any given size by extending the combinatorial setting of species of structures to poset matrices. Three of these partial composition operations are shown to define an operad structure for poset matrices. The structural properties of poset matrices and their duals are then studied based on their associated operad constructions.

Operad Structure of Poset Matrices

Abstract

This paper examines operad structures derived from poset matrices by formulating a set of new construction rules for poset matrices. In this direction, eleven different partial composition operations will be introduced as the basis for the construction of poset matrices of any given size by extending the combinatorial setting of species of structures to poset matrices. Three of these partial composition operations are shown to define an operad structure for poset matrices. The structural properties of poset matrices and their duals are then studied based on their associated operad constructions.
Paper Structure (4 sections, 18 theorems, 43 equations)

This paper contains 4 sections, 18 theorems, 43 equations.

Table of Contents

  1. Introduction
  2. Structural properties of operads from poset matrices
  3. Operad construction of $3\times 3$ and $4\times 4$ poset matrices
  4. Structural properties of operads from dual poset matrices

Key Result

Theorem 2.1

Let $A\in{\mathcal{P}M}(n)$,$B\in{\mathcal{P}M}(m),$ and let $C$ be the matrix defined as follows: In all cases $C$ is a poset matrix.

Theorems & Definitions (57)

  • Definition 1.1
  • Theorem 2.1
  • proof
  • Theorem 2.2
  • proof
  • Corollary 2.3
  • Lemma 2.4
  • proof
  • Example 2.5
  • Theorem 2.6
  • ...and 47 more