Motivated by some examples from functional programming, we propose a generalization of the notion of trace to symmetric premonoidal categories and of Conway operators to Freyd categories. We show that in a Freyd category, these notions are equivalent, generalizing a well-known theorem relating traces and Conway operators in Cartesian categories.

Algebraic systems of equations define functions using recursion where parameter passing is permitted. This generalizes the notion of a rational system of equations where parameter passing is prohibited. It has been known for some time that algebraic systems in Greibach Normal Form have unique solutions. This paper presents a categorical approach to algebraic systems of equations which generalizes the traditional approach in two ways i) we define algebraic equations for locally finitely presentable categories rather than just Set; and ii) we define algebraic equations to allow right-hand sides which need not consist of finite terms. We show these generalized algebraic systems of equations have unique solutions by replacing the traditional metric-theoretic arguments with coalgebraic arguments.

It is well known that, given an endofunctor H on a category C , the initial (A+H-)-algebras (if existing), i.e. , the algebras of (wellfounded) H-terms over different variable supplies A, give rise to a monad with substitution as the extension operation (the free monad induced by the functor H). Moss [17] and Aczel, Adámek, Milius and Velebil [12] have shown that a similar monad, which even enjoys the additional special property of having iterations for all guarded substitution rules (complete iterativeness), arises from the inverses of the final (A+H-)-coalgebras (if existing), i.e. , the algebras of non-wellfounded H-terms. We show that, upon an appropriate generalization of the notion of substitution, the same can more generally be said about the initial T'(A,-)-algebras resp. the inverses of the final T'(A,-)-coalgebras for any endobifunctor T' on any category C such that the functors T'(-,X) uniformly carry a monad structure.

The basic framework of domain μ-calculus was formulated in [39] more than ten years ago. This paper provides an improved formulation of a fragment of the μ-calculus without function space or powerdomain constructions, and studies some open problems related to this μ-calculus such as decidability and expressive power. A class of language equations is introduced for encoding μ-formulas in order to derive results related to decidability and expressive power of non-trivial fragments of the domain μ-calculus. The existence and uniqueness of solutions to this class of language equations constitute an important component of this approach. Our formulation is based on the recent work of Leiss [23], who established a sophisticated framework for solving language equations using Boolean automata (a.k.a. alternating automata [12,35]) and a generalized notion of language derivatives. Additionally, the early notion of even-linear grammars is adopted here to treat another fragment of the domain μ-calculus.

We investigate a Gentzen-style proof system for the first-order μ-calculus based on cyclic proofs, produced by unfolding fixed point formulas and detecting repeated proof goals. Our system uses explicit ordinal variables and approximations to support a simple semantic induction discharge condition which ensures the well-foundedness of inductive reasoning. As the main result of this paper we propose a new syntactic discharge condition based on traces and establish its equivalence with the semantic condition. We give an automata-theoretic reformulation of this condition which is more suitable for practical proofs. For a detailed comparison with previous work we consider two simpler syntactic conditions and show that they are more restrictive than our new condition.