RESEARCH STATEMENT My current research interests lie in ...

4 STEFAN MILIUS3.1. Iterative Algebras for a Base. One line of our present research has been inspired by the workof Uustalu [44]. He proposed to study complete iterativity with respect to a so-called base in lieu of anendofunctor. A base is a functor from the category A to the category of (finitary) monads on A. Bases allowan interesting extension of our theory which captures algebras satisfying certain equations and where theiterativity can be restricted. In fact, we have taken up the task to extend the results on iterative algebrasand iterative monads of [4, 6] to bases with the series of papers [8, 9, 10, 11, 12] most of which are still inpreparation.Research Problem 3.1. Investigate completely iterative algebras for a base and extend our results of [1, 34]to this setting.3.2. Algebraic Trees. An important application of the theory of iterativity for a base is presented in [13]:a category theoretic description of the monad of algebraic trees. Classically, algebraic trees over a signatureΣ are those trees arising as the uninterpreted solutions of all recursive program schemes with the givens fromΣ, see [18]. We have shown how to obtain the monad of algebraic trees from the free iterative monad for abase on the category of finitary endofunctors of Set.Research Problem 3.2.(i) Formulate and establish the following properties of algebraic trees in the category-theoretic setting:(a) algebraic trees form an iterative theory;(b) algebraic trees are closed under second-order substitution;(c) algebraic trees are rational trees over a signature with additional explicit substitution operations.(ii) Find a universal property that characterizes the monad of algebraic trees up to isomorphism.In contrast to all Σ-trees and all rational Σ-trees, respectively, a universal property is not known foralgebraic trees in the classical setting. So it would be a great success of the category-theoretic approach toprovide such a result.3.3. Iterative Reflection of a Monad. Our previous work shows how to obtain a free iterative monad ona finitary endofunctor of a locally finitely presentable category. How about starting with a finitary monad?Research Problem 3.3. Does there exist an iterative reflection of a given finitary monad of a locally finitelypresentable category?This would yield another important new result in the classical setting; an iterative theory free on (thetheory given by) a signature and a finite set of equations involving operation symbols of that signature.I have a partial result [36] showing that every ideal finitary monad of Set has an iterative reflection. Theconcept of an ideal theory was introduced by Elgot in order to study iterativity, see [19]. Ideal monads forma subcategory of all finitary monads of Set. Although the proof of the partial result provides a coalgebraicconstruction one does not obtain an explicit description of the iterative reflection from that. But sucha description would be highly desirable in order to obtain further examples of iterative theories and tounderstand whether the partial result can be extended beyond Set.Research Problem 3.4. Give an explicit description of an iterative reflection of an ideal finitary monadof Set.For free iterative monads on endofunctors of Set an explicit description is provided by Adámek and myselfin [3].Similar problems as above should also be addressed in the completely iterative case. Among expectedapplications of the theory developed to answer those problems are a semantics of recursive program schemesusing operations satisfying equations.3.4. More on Recursive Program Schemes. Our work on solutions of recursive program schemes wasjust the beginning of a long term project to rework algebraic semantics with category-theoretic methods.The next steps include:(a) Find other applications. For example, working with the power-set functor we obtain applicationsin the theory of hypersets of Barwise and Moss [15]. Other possible applications lie in the realm ofprocess calculi, e. g. Milner’s CCS [39].