Algebraic Signatures Enriched by Dependency Structure

Classical single-sorted algebraic signatures are defined as sets of operation symbols together with arities. In their many-sorted variant they also list sort symbols and use sort-sequences as operation types. An operation type not only indicates sorts of parameters, but also constitutes dependency between an operation and a set of sorts. In the paper we define algebraic signatures with dependency relation on their symbols. In modal logics theory, structures like 〈W,R〉, where W is a set and R ⊆ W ×W is a transitive relation, are called transitive Kripke frames [Seg70]. Part of our result is a definition of a construction of non-empty products in the category of transitive Kripke frames and p-morphisms. In general not all such products exist, but when the class of relations is restricted to bounded strict orders, the category lacks only the final object to be finitely (co)complete. Finally we define a category AlgSigDep of signatures with dependencies and we prove that it also has all finite (co)limits, with the exception of the final object.