Catoids and modal convolution algebras


We show how modal quantales arise as convolution algebras Q^X of functions from catoids X, that is, multisemigroups with a source map \ell and a target map r, into modal quantales Q, which can be seen as weight or value algebras. In the tradition of boolean algebras with operators we study modal correspondences between algebraic laws in X, Q and Q^X. The class of catoids we introduce generalises Schweizer and Sklar’s function systems and object-free categories to a setting isomorphic to algebras of ternary relations, as they are used for boolean algebras with operators and substructural logics. Our results provide a generic construction of weighted modal quantales from such multisemigroups. It is illustrated by many examples. We also discuss how these results generalise to a setting that supports reasoning with stochastic matrices or probabilistic predicate transformers.