Tukey morphisms between finite relations

TL;DR

This work investigates Tukey morphisms between binary relations, establishing several fundamental lemmas, and gives a construction of finite binary relations with arbitrary dominating number and dual dominating number.

Abstract

We investigate Tukey morphisms between binary relations, establishing several fundamental lemmas. We then specialize to finite binary relations, using computational methods to classify all binary relations with at most $6$ points in the domain and codomain up to bimorphism. Finally we give a construction of finite binary relations with arbitrary dominating number and dual dominating number.

Figures (8)

• Figure 1: A morphism from $\bm{A}$ to $\bm{B}$.
• Figure 2: The red ellipse denotes a minimal dominating family, the blue circle denotes a dominating family of minimum size.
• Figure 3: There relations $\bm{C}$ and $\bm{B}\oplus\bm{B'}$ are bimorphic, but $\bm{C}$ is not the disjoint union of nonempty subrelations.
• Figure 4: Finding the Skeleton Bimorphic Form of a Relation. Maximal points in $A_{+}$ are circled in red. Minimal points in $A_{-}$ are circled in blue.
• Figure 5: Hasse Diagram of skeleton forms of binary relations of order $\leq5$

Theorems & Definitions (34)

• Definition 1.1
• Proposition 1.2: Blass blass
• Definition 1.3
• Proposition 1.4: Blass blass
• Corollary 1.5
• Proposition 2.1
• Definition 2.2
• Lemma 2.3
• proof
• Corollary 2.4