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Diagonals and Block-Ordered Relations

Roland Backhouse, Ed Voermans

TL;DR

This work develops a point-free relation-algebra framework to connect block-ordered relations with the diagonal ΔR (the difunctional difference). It introduces core/index theory and residuals (factors) to replace nested complements, and proves that a relation is block-ordered precisely when its core is a provisional ordering, captured by the Analogie Frappante theorem. A key advance is showing that ΔJ is an index of ΔR whenever J is an index of R, and that block-orderings are characterized by ΔR through R< and R> relations. The results place block-ordered and staircase structures within a unifying Galois-connection perspective, enabling efficient representations and tests for block-ordering via cores and diagonals. The paper also clarifies historical claims about Ferrers relations, emphasizing a factor-based approach and correcting overreliance on complement-centered reasoning.

Abstract

More than 70 years ago, Jaques Riguet suggested the existence of an ``analogie frappante'' (striking analogy) between so-called ``relations de Ferrers'' and a class of difunctional relations, members of which we call ``diagonals''. Inspired by his suggestion, we formulate an ``analogie frappante'' linking the notion of a block-ordered relation and the notion of the diagonal of a relation. We formulate several novel properties of the core/index of a diagonal, and use these properties to rephrase our ``analogie frappante''. Loosely speaking, we show that a block-ordered relation is a provisional ordering up to isomorphism and reduction to its core. (Our theorems make this informal statement precise.) Unlike Riguet (and others who follow his example), we avoid almost entirely the use of nested complements to express and reason about properties of these notions: we use factors (aka residuals) instead. The only (and inevitable) exception to this is to show that our definition of a ``staircase'' relation is equivalent to Riguet's definition of a ``relation de Ferrers''. Our ``analogie frappante'' also makes it obvious that a ``staircase'' relation is not necessarily block-ordered, in spite of the mental picture of such a relation presented by Riguet.

Diagonals and Block-Ordered Relations

TL;DR

This work develops a point-free relation-algebra framework to connect block-ordered relations with the diagonal ΔR (the difunctional difference). It introduces core/index theory and residuals (factors) to replace nested complements, and proves that a relation is block-ordered precisely when its core is a provisional ordering, captured by the Analogie Frappante theorem. A key advance is showing that ΔJ is an index of ΔR whenever J is an index of R, and that block-orderings are characterized by ΔR through R< and R> relations. The results place block-ordered and staircase structures within a unifying Galois-connection perspective, enabling efficient representations and tests for block-ordering via cores and diagonals. The paper also clarifies historical claims about Ferrers relations, emphasizing a factor-based approach and correcting overreliance on complement-centered reasoning.

Abstract

More than 70 years ago, Jaques Riguet suggested the existence of an ``analogie frappante'' (striking analogy) between so-called ``relations de Ferrers'' and a class of difunctional relations, members of which we call ``diagonals''. Inspired by his suggestion, we formulate an ``analogie frappante'' linking the notion of a block-ordered relation and the notion of the diagonal of a relation. We formulate several novel properties of the core/index of a diagonal, and use these properties to rephrase our ``analogie frappante''. Loosely speaking, we show that a block-ordered relation is a provisional ordering up to isomorphism and reduction to its core. (Our theorems make this informal statement precise.) Unlike Riguet (and others who follow his example), we avoid almost entirely the use of nested complements to express and reason about properties of these notions: we use factors (aka residuals) instead. The only (and inevitable) exception to this is to show that our definition of a ``staircase'' relation is equivalent to Riguet's definition of a ``relation de Ferrers''. Our ``analogie frappante'' also makes it obvious that a ``staircase'' relation is not necessarily block-ordered, in spite of the mental picture of such a relation presented by Riguet.
Paper Structure (26 sections, 42 theorems, 143 equations, 5 figures)

This paper contains 26 sections, 42 theorems, 143 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Axioms of Point-free Relation Algebra
  3. Summary
  4. Factors
  5. Some Definitions
  6. Coreflexives
  7. The Domain Operators
  8. Pers and Per Domains
  9. Functionality
  10. Difunctions
  11. Provisional Orderings
  12. Squares and Rectangles
  13. Isomorphic Relations
  14. Indexes and Core Relations
  15. Definitions
  16. ...and 11 more sections

Key Result

Lemma 12

For all $R$ and $S$, Symmetrically, Also,

Figures (5)

  • Figure 1: Mental Picture of a Staircase Relation
  • Figure 2: A Provisional Preorder
  • Figure 3: Diagonal of an Instance of the Membership Relation
  • Figure 4: A Relation on Two Posets
  • Figure 5: A Block-Ordered Membership Relation

Theorems & Definitions (68)

  • Lemma 12
  • Definition 16: Domain Operators
  • Definition 22: Right and Left Per Domains
  • Lemma 25
  • Definition 26: Partial Equivalence Relation (per)
  • Lemma 33
  • Lemma 34
  • Lemma 35
  • Lemma 36
  • Lemma 37
  • ...and 58 more