The order structure of real spectra
by
Francois Lucas
University of Angers
Coauthors: Maximo Dickmann (Paris VII) and Daniel Gluschankof

Ever since the Real Spectrum of a ring was introduced by Coste and Coste-Roy at the end of the 1970's, it has been an outstanding problem to decide wether every spectral space whose order of specialisation is a root sysem can be realized as the real spectrum of a commutative unitary ring. In 1994 Delzell and Madden gave a counterexample. Here we deal with a purely order theoretic aspect of the problem showing that any jump-dense, complete root system is isomorphic to the real spectrum of a commutative unitary ring. Along the way we produce several classes of rings with natural finiteness properties and other pleasant features such as a lattice structure in which every positive element has a square root.

Date received: April 29, 1999