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Poset Matrix Structure Via Partial Composition Operations

Arnauld Mesinga Mwafise

TL;DR

The structure of poset matrices is examined by formulating a set of new construction rules for this purpose by extending the combinatorial setting of species of structures to posetMatrices by introducing the technique of partial composition operations.

Abstract

This paper examines the structure of poset matrices by formulating a set of new construction rules for this purpose. In this direction, the technique of partial composition operation 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. More specifically, three new partial composition operations that apply to poset matrices are defined as the foundation for this study. Several new structural properties derived from viewing any poset matrix and its dual in terms of these operations are highlighted.

Poset Matrix Structure Via Partial Composition Operations

TL;DR

The structure of poset matrices is examined by formulating a set of new construction rules for this purpose by extending the combinatorial setting of species of structures to posetMatrices by introducing the technique of partial composition operations.

Abstract

This paper examines the structure of poset matrices by formulating a set of new construction rules for this purpose. In this direction, the technique of partial composition operation 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. More specifically, three new partial composition operations that apply to poset matrices are defined as the foundation for this study. Several new structural properties derived from viewing any poset matrix and its dual in terms of these operations are highlighted.
Paper Structure (3 sections, 9 theorems, 12 equations)

This paper contains 3 sections, 9 theorems, 12 equations.

Table of Contents

  1. Introduction
  2. Partial composition operations of poset matrices
  3. Some key structural properties of poset matrices using their partial composition operations

Key Result

Theorem 2.1

Let $A=[a_{i,j}]$ be a poset matrix of size $n$ and let $L$ be any $n$-element labeling set for the row and column indices of $A.$ Then whenever $i\neq j$,

Theorems & Definitions (39)

  • Theorem 2.1
  • Remark 2.2
  • Example 2.3
  • Example 2.4
  • Theorem 2.5
  • Theorem 2.6
  • Definition 3.1
  • Theorem 3.2
  • Theorem 3.3
  • Theorem 3.4
  • ...and 29 more