Abstract
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→zA 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 ZId(L(A)) of the z-ideals of the lattice L(A).
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