Sebastian Kerkhoff

A GENERAL DUALITY THEORY FOR CLONES

Inspired by work of Masulovic, we outline a general duality theory for clones that will allow us to dualize any given clone, together with its relational counterpart and the relationship between them. Afterwards, we put the approach to work and illustrate it by producing some specific results for concrete examples as well as some general results that come from studying the duals of clones in a rather abstract fashion.

A Short Introduction to Clones

In universal algebra, clones are used to study algebras abstracted from their signature. The aim of this paper is to give a brief introduction to the theory thereof. We give basic definitions and examples, and we present several results and open problems, selected from almost one hundred years of ongoing research. We also discuss what is arguably the most important tool to study clones - the Galois connection between operations and relations built on the notion of preservation. We conclude the paper by explaining the connection between clones and the closely related category theoretic notion of Lawvere theory.

A connection between clone theory and FCA provided by duality theory

The aim of this paper is to show how Formal Concept Analysis can be used for the benefit of clone theory. More precisely, we show how a recently developed duality theory for clones can be used to dualize clones over bounded lattices into the framework of Formal Concept Analysis, where they can be investigated with techniques very different from those that universal algebraists are usually armed with. We also illustrate this approach with some small examples.

Dualizing Clones as Models of Lawvere Theories

While universal algebraists are well aware of the equivalence between abstract clones and Lawvere theories as well as that of concrete clones and models of Lawvere theories in the category of sets, they almost never use the category-theoretic framework. It seems as if they simply do not see a reason why it might be beneficial to use category theory in order to study the problems that they are interested in. In this paper, we argue that the possibility of applying duality theory might be such a reason, and we support this claim by outlining how treating and dualizing clones as models of Lawvere theories can be beneficial for the classical problem of studying the lattice of clones on a given set. In particular, we give several examples of concrete results that are obtained with this method.

LOCALLY COMPACT GROUPS ADMITTING FAITHFUL STRONGLY CHAOTIC ACTIONS ON HAUSDORFF SPACES

In this paper we provide a geometric characterization of those locally compact Hausdorff topological groups which admit a faithful strongly chaotic continuous action on some Hausdorff space.

Dynamical Systems in Categories

In this article we establish a bridge between dynamical systems, including topological and measurable dynamical systems as well as continuous skew product flows and nonautonomous dynamical systems; and coalgebras in categories having all finite products. We introduce a straightforward unifying definition of abstract dynamical system on finite product categories. Furthermore, we prove that such systems are in a unique correspondence with monadic algebras whose signature functor takes products with the time space. We discuss that the categories of topological spaces, metrisable and uniformisable spaces have exponential objects w.r.t. locally compact Hausdorff, strongly σ-compact or arbitrary time spaces as exponents, respectively. Exploiting the adjunction between taking products and exponential objects, we demonstrate a one-to-one correspondence between monadic algebras (given by dynamical systems) for the left-adjoint functor and comonadic coalgebras for the other. This, finally, provides a new, alternative perspective on dynamical systems.

Clones of Partial Cofunctions

We introduce and study clones of partial cofunctions on sets of arbitrary cardinality. In the process, we outline a general Galois theory similar to Pol-Inv, show some elementary results about the essential arity of clones of partial cofunctions, take a closer look at partial idempotent cofunctions, characterize all minimal clones of partial cofunctions, and show that the join of all minimal clones is the full clone (provided that the Axiom of Choice is assumed).

Dualities Induced by Topological Semirings

In this paper we establish a general duality theorem for compact Hausdorff spaces being recognizable over certain pairs consisting of a commutative unital topological semiring and a closed proper prime ideal. Indeed, we utilize the concept of blueprints and their localization to prove that the category of compact Hausdorff spaces generated by such a pair can be dually embedded into the category of commutative unital semirings if the pair possesses sufficiently many covering polynomials.

Concrete Dualities and Essential Arities

Many dualities arise in the same way: they are induced by dualizing objects. We show that these dualities are connected to a question occurring in universal algebra. Indeed, they cause a strong interplay between the essential arity of finitary operations in one category and the concrete form of the copowers in the other. We elaborate on this connection and its usefulness for universal algebra and clone theory in particular. As the papers main result we show that, under some mild assumptions, the essential arity of finitary operations from an object A to a finite object B in one category is bounded if and only if the concrete form of the copowers of the dual of A has a certain (easily verifiable) set-theoretic property.

Directed Tree Decompositions

In the problem session of the ICFCA 2006, Sandor Radeleczki asked for the meaning of the smallest integer k such that a given poset can be decomposed as the union of k directed trees. The problem also asks for the connection of this number to the order dimension. Since it was left open what kind of decomposition might be used, there is more than one reading of this problem. In the paper, we discuss different versions and give some answers to this open problem.