Arrow Operations in Categories of Lattice-valued Relations

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.