Mutually complementary partial orders

Mutually complementary partial orders

Two partial orders P = (X, S) and Q = (X, s’) are complementary if P fl Q = {(x, x): x E x} and the transitive closure of P U Q is {(x. y): x, y E X}. We investigate here the size w(n) of the largest set of pairwise complementary par!iai orders on a set of size n. In particular, for large n we construct L?(n/iogrt) mutually complementary partial orders of order n, and show on the other hand tha...

Families of Mutually Complementary Topologies

Several lattices of topologies on an infinite set are considered and bounds are given for the sup of the set of cardinals d such that there is a family of d mutually complementary topologies. Large classes of N<rtopologies are shown to have No-complements, and an example is given to prove that complementation is not, in general, a very selective topological operation.

Mutually Recursive Partial Functions

We provide a wrapper around the partial-function command which supports mutual recursion. Our results have been used to simplify the development of mutually recursive parsers, e.g., a parser to convert external proofs given in XML into some mutually recursive datatype within Isabelle/HOL.

Partial Orders on Partial Isometries

This paper studies three natural pre-orders of increasing generality on the set of all completely non-unitary partial isometries with equal defect indices. We show that the problem of determining when one partial isometry is less than another with respect to these pre-orders is equivalent to the existence of a bounded (or isometric) multiplier between two natural reproducing kernel Hilbert spac...

Extending Partial Orders to Dense Linear Orders