Prime elements from distributive quantales and radicals
by
Jan Paseka
Dept. of Mathematics, Faculty of Science, Masaryk University Brno

This work is intended as a step towards the development of the non-commutative topology using the approach of the theory of (involutive) quantales developed by C.J. Mulvey, J.W. Pelletier and J. Rosický and others. Quantales are certain partially ordered algebraic structures which generalize frames (pointless topologies) as well as various lattices of multiplicative ideals from ring theory and functional analysis (C*-algebras, von Neumann algebras). The relations on a set, under the operation of relational composition, also form an involutive quantale.

It is well known that 2-sided m-continuous quantales are spatial and that algebraic distributive quantales are spatial as well.

In this lecture we shall show that continuous distributive quantales are spatial. As a corollary, we get that each algebraic (stably continuous) distributive quantale is spatial. A Separation Lemma for Distributive Quantales is formulated. In the second part of this lecture we will establish the connection with the theory od radicals for algebraic distributive quantales.

Date received: February 26, 2003