Catoids and modal convolution algebras
In Algebra Universalis
Abstract 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.