Classification of finite spectral triples

It is known that the spin structure on Riemannian manifold can be extended to noncommutative geometry using the notion of spectral triple. For finite geometries, the corresponding finite spectral triples are completely described in terms of matrices and classified using diagrams. When tensorized with the ordinary space-time geometry, finite spectral triples give rise to Yang-Mills theories with spontaneous symmetry breaking, whose characteristic features are given within the diagrammatic approach: vertices of the diagram correspond to gauge multiplets of chiral fermions and links to Yukawa couplings. (C) 1998 Published by Elsevier Science B.V. All rights reserved.