Arrow Operations in Categories of Lattice-valued Relations
2026-08-17 • Discrete Mathematics
Discrete Mathematics
AI summaryⓘ
The authors study arrow allegories, which are abstract tools used to handle relations that have truth values from a specific logical structure called a Heyting algebra. Typically, these arrow allegories use the same truth value system for all relations, but the authors expand this to allow different truth value systems depending on the objects involved or even for each pair in the relation. They define three versions of these allegories to represent these scenarios and explore their properties using category theory. Their work clarifies how these extended structures behave mathematically.
Arrow allegoriesHeyting algebraLattice-valued relationsCategory theoryRelational structuresFull suballegoryArrow categoryTruth values
Authors
Fatemeh Jowkar, Michael Winter
Abstract
Arrow allegories provide a convenient abstract framework to work with lattice-valued relations, or more precisely, relations that use the elements of a given Heyting algebra as truth values. One characteristic of arrow allegories is that all relations of the given arrow allegory use the same Heyting algebra ${\mathcal H}$. In this paper we want to extend this approach to allegories where relations between different objects may use different lattices of truth values and even further to relations that use a different lattice of truth values for every pair in the relation. Therefore, we define three concrete allegories, $\mathrm{Rel}({\mathcal H})$, $\mathrm{Rel}^u({\mathcal H})$ and ${\mathcal H}{\rm-Rel}$, where the allegory listed later is a full suballegory of the previous ones. These three allegories capture the three different situations mentioned above. In particular, ${\mathcal H}{\rm-Rel}$ is the standard example of an arrow category. We investigate these allegories and provide suitable categorical definitions for these structures.