Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Isolated Suborders and their Application to Counting Closure Operators

Roland Glück

TL;DR

This work tackles counting closure operators on general ordered sets by introducing isolated suborders and quotient constructions $S/\sim_{S'}$ as a structural reduction tool. It develops a cryptomorphic framework linking closures and closure systems, and leverages isolated suborders to derive exact relations between closures on the original order and closures on quotients, enabling a recursive counting approach. The paper provides concrete formulas for base shapes (e.g., chains, diamonds) and establishes recursion rules for bottleneck and summit isolated suborders, culminating in an algorithmic scheme that decomposes complex orders into simpler components. While the approach is a generalization of lattice-focused results and relies on heuristic analysis for its complexity, it offers a principled path toward counting closures in orders that admit suitable isolated suborder structures, with practical implications for order theory and related computational problems.$S/ obreakspace\sim_{S'}$ and $|\mathcal{C}(S)|$ are central notation throughout.

Abstract

In this paper we investigate the interplay between isolated suborders and closures. Isolated suborders are a special kind of suborders and can be used to diminish the number of elements of an ordered set by means of a quotient construction. The decisive point is that there are simple formulae establishing relationships between the number of closures in the original ordered set and the quotient thereof induced by isolated suborders. We show how these connections can be used to derive a recursive algorithm for counting closures, provided the ordered set under consideration contains suitable isolated suborders.

Isolated Suborders and their Application to Counting Closure Operators

TL;DR

This work tackles counting closure operators on general ordered sets by introducing isolated suborders and quotient constructions as a structural reduction tool. It develops a cryptomorphic framework linking closures and closure systems, and leverages isolated suborders to derive exact relations between closures on the original order and closures on quotients, enabling a recursive counting approach. The paper provides concrete formulas for base shapes (e.g., chains, diamonds) and establishes recursion rules for bottleneck and summit isolated suborders, culminating in an algorithmic scheme that decomposes complex orders into simpler components. While the approach is a generalization of lattice-focused results and relies on heuristic analysis for its complexity, it offers a principled path toward counting closures in orders that admit suitable isolated suborder structures, with practical implications for order theory and related computational problems. and are central notation throughout.

Abstract

In this paper we investigate the interplay between isolated suborders and closures. Isolated suborders are a special kind of suborders and can be used to diminish the number of elements of an ordered set by means of a quotient construction. The decisive point is that there are simple formulae establishing relationships between the number of closures in the original ordered set and the quotient thereof induced by isolated suborders. We show how these connections can be used to derive a recursive algorithm for counting closures, provided the ordered set under consideration contains suitable isolated suborders.
Paper Structure (12 sections, 29 theorems, 5 equations, 2 figures, 2 algorithms)

This paper contains 12 sections, 29 theorems, 5 equations, 2 figures, 2 algorithms.

Table of Contents

  1. Introduction
  2. Basic Notions and Properties
  3. Closures
  4. Isolated Suborders
  5. Isolated Suborders and Closure Systems
  6. Computing Isolated Suborders
  7. Isolated Suborders and Separators
  8. Computing Isolated Suborders
  9. Counting Closure Operators
  10. Special Cases
  11. Counting Closures using Isolated Suborders
  12. Conclusion and Outlook

Key Result

Corollary 3.2

Let $c$ be a closure operator and $x,y\in S$ elements such that $y\geq c(x)$ ($y=c(x)$) holds. Then $x\leq y$ holds.

Figures (2)

  • Figure 1: Various Kinds of Isolated Suborders
  • Figure 2: A preclosure system on an isolated suborder (left), a closure system on a quotient (middle) and no closure system on the original order (right)

Theorems & Definitions (64)

  • Definition 3.1
  • Corollary 3.2
  • proof
  • Definition 3.3
  • Remark 3.4
  • Lemma 3.5
  • proof
  • Lemma 3.6
  • proof
  • Definition 4.1
  • ...and 54 more