Zipped coherent quantales

Document Type : Original Article


University of Bucharest


The aim of this paper is to define an abstract quantale framework for extending some properties of the zip rings (studied by Faith, Zelmanowitz, etc.) and the weak zip rings (defined by Ouyang). By taking as prototype the quantale of ideals of a zip ring (resp. a weak zip ring) we introduce the notion of zipped quantale (resp. weakly zipped quantale). The zipped quantales also generalize the zipped frames, defined by Dube and Blose in a recent paper. We define the zip (bounded  distributive) lattices and we prove that a coherent quantale A is weakly zipped iff the reticulation L(A) of A is a zip lattice.  From this result we obtain the following corollary: the coherent quantale A is weakly zipped iff the frame R(A) of the  radical elements of A is zipped. Such theorems allow us to extend to quantale framework a lot of results obtained by  Dube and Blose for the zipped frames and for the weak zip rings. 


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