Involutive and relational quantaloids (Q1592145)

The goal of this paper is to generalize results and constructions for involutive quantales (studied by Mulvey and Pelletier), as well as those for relational quantales (due to Brown and Gurr), to the level of quantaloids. A quantaloid is a locally small category enriched in the autonomous category \({\mathcal S}{\mathcal L}\) of sup-lattices. One-object quantaloids are precisely unital quantales. For an overview on quantaloids, see the reviewer's book [\textit{K. I. Rosenthal}, ``The theory of quantaloids'', Pitman Res. Notes Math. 348 (1996; Zbl 0845.18003)]. The first part of the paper develops the theory of involutive quantaloids and derives various properties for them under some mild additional conditions. Connections with Freyd's law of modularity are also discussed. The second part of the paper proves an important representation theorem for arbitrary quantaloids. One of the fundamental examples of a quantaloid is that of \(\text{Rel}(X)\), relations on a set \(X\) under relational composition. Starting with a quantaloid \(Q\) and a generating family \(G\) for it, the author constructs a new quantaloid \(\text{Rel} (G)\) of ordered relations and proves that \(Q\) is in fact isomorphic to \(\text{Rel} (G)\). This generalizes a similar representation theorem for quantales proved by \textit{C. Brown} and \textit{D. Gurr} [``A representation theorem for quantales'', J. Pure Appl. Algebra 85, No. 1, 27-42 (1993; Zbl 0776.06011)].