Sorting algorithms can be classiied in many diierent ways. The way presented here is by expressing the algorithms as functional programs and to classify them by means of their recursion patterns. These patterns on their turn can be classiied as the natural recursion patterns that destruct or construct a given data-type, the so called cata-and anamorphisms respectively. We show that the selectio...

The aim is the theorems of the title and the corollary that the tensor product of two free crossed resolutions of groups or groupoids is also a free crossed resolution of the product group or groupoid. The route to this corollary is through the equivalence of the category of crossed complexes with that of cubical ω-groupoids with connections where the initial definition of the tensor product li...

Given a map f in the category ω-Cpo of ω-complete posets, exponentiability of f in ω-Cpo easily implies exponentiability of f in the category Pos of posets, while the converse is not true. We find then the extra conditions needed on f exponentiable in Pos to be exponentiable in ω-Cpo, showing the existence of partial products of the two-point ordered set S = {0 < 1} (Theorem 1.8). Using this ch...

We investigate different notions of recognizability for a free monoid morphism σ : A∗ → B∗. Full recognizability occurs when each (aperiodic) point in BZ admits at most one tiling with words σ(a), a ∈ A. This is stronger than the classical notion of recognizability of a substitution σ : A∗ → A∗, where the tiling must be compatible with the language of the substitution. We show that if |A| = 2, ...