Work

Algebraically universal categories of relational structures

Year: 2025

Type: article

Abstract: We study the problem of representability of categories of algebras in categories of relational structures. This is a general framework for a long line of research pertaining to the realisation of algebraic structures in graphs. Drawing inspiration from a combinatorial property of classes of finite graphs known as nowhere density that originates from the work of Nešetřil and Ossona de Mendez, we establish a partial characterisation of those relational categories which are algebraically universal, meaning that they fully embed all categories of algebras. More precisely, we show that the any algebraically universal category of relational structures must necessarily contain subdivided complete graphs of any infinite size. Conversely, we establish that any relational category closed under removal of relations and having this property may be oriented to obtain an algebraically universal category. For the proof of the above, we develop a categorical framework for relational gadget constructions. This generalises existing work on algebraic representability in categories of finite graphs to categories of relational structures of unbounded size.

Source: Journal of Pure and Applied Algebra

DOI: 10.1016/j.jpaa.2025.107984

Author: Ioannis Eleftheriadis

Institution: University of Cambridge

Language: English

Topic: Advanced Topology and Set Theory

Subfield: Geometry and Topology

Field: Mathematics

Domain: Physical Sciences

Open Access status: hybrid

Estimated APC paid: $3,550

Funders: Cambridge Trust, Alexander S. Onassis Public Benefit Foundation, Gates Cambridge Trust