Continuously extending partial functions

Extending partial combinatory algebras

We give a negative answer to the question whether every partial combinatory algebra can be completed. The explicit counterexample will be an intricately constructed term model, the construction and the proof that it works heavily depending on syntactic techniques. In particular, it is a nice example of reasoning with elementary diagrams and descendants. We also include a domain-theoretic proof ...

Extending Partial Projective Planes

If a finite partial projective plane can be extended to a projective plane of order n, then it can be extended by a sequence of one line extensions. We describe necessary conditions for such an extension to exist, and restrictions which can be imposed on the intermediate sequence of extensions. This method can be used to attempt to construct a projective plane of non-prime-power order. 1. Parti...

Extending partial combinatory algebras

Extending partial tournaments

Let A be a (0, 1, ∗)-matrix with main diagonal all 0’s and such that if ai,j = 1 or ∗ then aj,i = ∗ or 0. Underwhat conditions on the row sums, and or column sums, of A is it possible to change the ∗’s to 0’s or 1’s and obtain a tournament matrix (the adjacency matrix of a tournament) with a specified score sequence? We answer this question in the case of regular and nearly regular tournaments....

Extending partial permtations

This document describes a simple problem about extending partial permutations which may be recursively unsolvable, and discusses the context.