Monads on Categories of Relational Structures

We introduce a framework for universal algebra in categories of relational structures given by finitary relational signatures and finitary or infinitary Horn theories, with the arity ? of a Horn theory understood as a strict upper bound on the number of premisses in its axioms; key examples include partial orders (? = ?) or metric spaces (? = ??). We establish a bijective correspondence between ?-accessible enriched monads on the given category of relational structures and a notion of ?-ary algebraic theories (i.e. with operations of arity < ?), with the syntax of algebraic theories induced by the relational signature (e.g. inequations or equations-up-to-?). We provide a generic sound and complete derivation system for such relational algebraic theories, thus in particular recovering (extensions of) recent systems of this type for monads on partial orders and metric spaces by instantiation. In particular, we present an ??-ary algebraic theory of metric completion. The theory-to-monad direction of our correspondence remains true for the case of ?-ary algebraic theories and ?-accessible monads for ? < ?, e.g. for finitary theories over metric spaces

Supported Sets - A New Foundation for Nominal Sets and Automata

The present work proposes and discusses the category of supported sets which provides a uniform foundation for nominal sets of various kinds, such as those for equality symmetry, for the order symmetry, and renaming sets. We show that all these differently flavoured categories of nominal sets are monadic over supported sets. Thus, supported sets provide a canonical finite way to represent nominal sets and the automata therein, e.g. register automata and coalgebras in general. Name binding in supported sets is modelled by a functor following the idea of de Bruijn indices. This functor lifts to the well-known abstraction functor in nominal sets. Together with the monadicity result, this gives rise to a transformation process from finite coalgebras in supported sets to orbit-finite coalgebras in nominal sets. One instance of this process transforms the finite representation of a register automaton in supported sets into its configuration automaton in nominal sets

Proper Functors and Fixed Points for Finite Behaviour

The rational fixed point of a set functor is well-known to capture the behaviour of finite coalgebras. In this paper we consider functors on algebraic categories. For them the rational fixed point may no longer be fully abstract, i.e. a subcoalgebra of the final coalgebra. Inspired by \'Esik and Maletti's notion of a proper semiring, we introduce the notion of a proper functor. We show that for proper functors the rational fixed point is determined as the colimit of all coalgebras with a free finitely generated algebra as carrier and it is a subcoalgebra of the final coalgebra. Moreover, we prove that a functor is proper if and only if that colimit is a subcoalgebra of the final coalgebra. These results serve as technical tools for soundness and completeness proofs for coalgebraic regular expression calculi, e.g. for weighted automata