[Paper Review] Universal Associative Geometry

Wolfgang Bertram|arXiv (Cornell University)|Jun 6, 2014

Advanced Topics in Algebra10 references3 citations

TL;DR

This paper introduces a universal framework for associative geometry by extending associoid structures—such as pregroupoids and principal equivalence relations—from a set Ω to its power set 𝒫(Ω), using relational composition under commuting equivalence relations. The key contribution is a duality between subsets and equivalence relations, generalizing groupoid structures and bisections via ternary operations, with applications to torsors and homogeneous geometries.

ABSTRACT

We generalize parts of the theory of associative geometries developed by Kinyon and the author in the framework of universal algebra: we prove that certain associoid structures, such as pregroupoids and principal equivalence relations, have a natural prolongation from a set to its the power set. We reinvestigate the case of homogeneous pregroupoids (corresponding to the projective geometry of a group) from the point of view of pairs of commuting principal equivalence relations. We use the ternary approach to groupoids developed by Anders Kock, and the torsors defined by our construction can be seen as a generalisation of the known groups of bisections of a groupoid.

Motivation & Objective

• To develop a universal algebraic framework for associative geometry beyond specific instances.
• To generalize the duality between subsets and equivalence relations in a way that completes the incomplete duality noted in universal algebra.
• To show that pregroupoids and principal equivalence relations naturally extend from Ω to 𝒫(Ω) under commuting equivalence relations.
• To reformulate groupoid and torsor structures using ternary operations and bisections, unifying their algebraic and geometric aspects.
• To establish a canonical construction of semi-torsors and torsors from commuting pairs of equivalence relations.

Proposed method

• Using Kock’s ternary product formalism, where the product is written as [xyz] or xy⁻¹z, to define associative and para-associative structures.
• Defining the domain of the ternary product via two equivalence relations a and b on a set M, forming a pregroupoid when the product is partially defined.
• Constructing a relational composition x∘y⁻¹∘z on subsets of Ω by fixing a pair of commuting equivalence relations (a,b), enabling 'composition of sets' via structure.
• Introducing the concept of a bisection s in a pregroupoid (M,a,b), which allows defining inversion and composition via [g a_g b_g] and [g b_g h], respectively.
• Proving that a pregroupoid with a fixed bisection s becomes a groupoid with s as the set of units, generalizing the known correspondence between groupoids and pregroupoids.
• Using the canonical kernel and associative pair structure (U_a, U_b) to analyze the behavior of the ternary product and its prolongation to power sets.

Experimental results

Research questions

• RQ1 How can the notion of relational composition be extended from relations to subsets of a set Ω, given a pair of commuting equivalence relations?
• RQ2 What is the universal algebraic structure that generalizes both groups and groupoids via a single ternary operation?
• RQ3 Can the duality between subsets and equivalence relations be completed in a way that allows 'composing' subsets using relational structure?
• RQ4 How do pregroupoids and principal equivalence relations extend from a set Ω to its power set 𝒫(Ω)?
• RQ5 What is the role of commuting pairs of equivalence relations in defining semi-torsors and torsors in the universal associative geometry framework?

Key findings

• The paper establishes that for any pair of commuting equivalence relations (a,b) on a set Ω, the power set 𝒫(Ω) inherits a natural structure of a semi-torsor via relational composition x∘y⁻¹∘z.
• A pregroupoid (M,a,b) with a fixed bisection s becomes a groupoid with s as the set of units, and the ternary product [xyz] encodes both multiplication and inversion.
• The construction of torsors from pregroupoids generalizes the known groups of bisections of a groupoid, extending them to a universal algebraic setting.
• The canonical kernel and associative pair (U_a, U_b) provide a structural framework for analyzing the prolongation of associoid structures to power sets.
• In the homogeneous case, the structure of (U_a, U_b) corresponds to a dual pair of torsors, generalizing the duality between a vector space and its dual.
• The theory provides a universal geometric language for associative algebras, analogous to how Lie groups relate to Lie algebras, but in the context of universal algebra and relational composition.