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.