Continuous and Discrete Geometric Spaces as Partial Vector Spaces
by
Julian Webster
Imperial College, U. of London

We introduce partial vector spaces (PVS) as `vector spaces' in which the scalars are partial rather than total functions. Z<sup>n</sup> may be regarded as a PVS, and the theory is meant in part to be an axiomatic approach to discrete geometry. The main points of the theory are: (1) PVS have substantial geometric structure, and the basic theory is quite similar to basic linear algebra; (2) There exists a free vector space over any partial vector space (R<sup>n</sup> is the free vector space over Z<sup>n</sup>), which provides a close and formal connection between `discrete' and `continuous' geometric spaces.

Date received: June 17, 1999