mz-elements in coherent quantales.

In this paper we define and study the mz -elements of an algebraic quantale as an abstraction of the mz -ideals of a commutative ring, recently introduced by Ighedo and McGovern. Using a result of Banaschewski, we prove that the set zA of the mz -elements of a coherent quantale A is a coherent frame, as the image of a localic nucleus s : A → A. We show that s : A → z A is a codense coherent quantale morphism, then we use the morphism s in order to obtain the quantale generalizations of some results obtained by Ighedo and McGovern. We study the relationship between the mz -elements of a coherent quantale A and the z -elements of the reticulation L (A) of A. In particular, we prove that the frame zA is isomorphic with the frame Z I d (L (A)) of the z -ideals of the lattice L (A). [ABSTRACT FROM AUTHOR]