Zipped coherent quantales

Document Type : Original Article

Author

University of Bucharest

Abstract

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. 

Keywords


[1] M.F. Atiyah, I.G. MacDonald, Introduction to commutative algebra, Addison-Wesley Publ. Comp., 1969.
[2] R. Balbes, Ph. Dwinger, Distributive lattices, Univversity of Missouri Press, 1974.
[3] B. Banaschewski, Gelfand and exchange rings: Their spectra in point free topology, The Arabian Journal for Science and Engineering, 25(2C) (2000), 3–22.
[4] B. Banaschewski, A. Pultr, Booleanization, Cahiers de Topologie et Géométrie Différentielle Catégoriques, 37 (1996), 41–60.
[5] D. Cheptea, G. Georgescu, Boolean lifting property in quantales, Soft Computing, 24 (2020), 6169–6181.
[6] M. Dickmann, N. Schwarz, M. Tressl, Spectral spaces, Cambridge University Press, 2019.
[7] T. Dube, S. Blose, Algebraic frames in which dense elements are above dense compact elements, Algebra Universalis, 88(3) (2023). DOI:10.1007/s00012-022-00799-w.
[8] P. Eklund, J.G. Garcia, U. Hohle, J. Kortelainen, Semigroups in complete lattices: Quantales, modules and related topics, Springer, 2018.
[9] A. Facchini, C.A. Pinocchiaro, G. Janelidze, Abstractly constructed prime spectra, Algebra Universalis, 83(8) (2022). DOI:10.107/s00012-021-00764-z.
[10] C. Faith, Rings with zero intersection property: Zipp rings, Publicationes Mathematicae, 33 (1989), 329–338.
[11] C. Faith, Annihilator ideals, associate primes and Kasch-McCoy commutative rings, Communications in Algebra, 19 (1991), 1867–1892.
[12] N. Galatos, P. Jipsen, T. Kowalski, H. Ono, Residuated lattices: An algebraic glimpse at structural logics, Studies in Logic and The Foundation of Mathematics, 151, Elsevier, 2007.
[13] G. Georgescu, The reticulation of a quantale, Revue Roumaine de Mathématique Pures et Appliquées, 40(7-8) (1995), 619–631.
[14] G. Georgescu, Some classes of quantale morphisms, Journal of Algebra, Number Theory, Advances and Applications, 24(2) (2021), 111–153.
[15] G. Georgescu, Flat topology on the spectra of quantales, Fuzzy Sets and Systems, 406 (2021), 22–41.
[16] M. Hochster, Prime ideals structures in commutative rings, Transactions of the American Mathematical Society, 142 (1969), 43–60.
[17] P. Jipsen, Generalization of Boolean products for lattice-ordered algebras, Annals of Pure and Applied Logic, 161 (2009), 224–234.
[18] P.T. Johnstone, Stone spaces, Cambridge University Press, 1982.
[19] J. Martinez, Abstract ideal theory, Ordered Algebraic Structures, Lecture Notes in Pure and Applied Mathematics, 99 Marcel Dekker, New York, 1985.
[20] J. Martinez, An innocent theorem of Banaschewski, applied to an unsuspecting theorem of De Marco, and the aftermath thereof, Forum Mathematicum, 25 (2013), 565–596.
[21] L. Ouyang, Ore extension of zip rings, Glasgow Mathematical Journal, 51 (2009), 525–537.
[22] L. Ouyang, G.F. Birkenmeier, Weak annihilators over extension rings, Bulletin of the Malaysian Mathematical Sciences Society, 35(2) (2012), 345–347.
[23] J. Paseka, Regular and normal quantales, Archiv der Mathematik (Brno), 22 (1996), 203–210.
[24] J. Paseka, J. Rosicky, Quantales, current research in operational quantum logic: Algebras, categories and languages, Foundations of Physics, 111 (2000), 245–262.
[25] J. Picado, A. Pultr, Frames and locales: Topology without points, Frontiers in Mathematics, Springer, Basel, 2012. [26] K.I. Rosenthal, Quantales and their applications, Longman Scientific and Technical, 1989.
[27] H. Simmons, Reticulated rings, Journal of Algebra, 66 (1980), 169–192.
[28] J.M. Zelmanowitz, The finite intersection property on annihilator right ideals, Proceedings of the American Mathematical Society, 57 (1976), 213–216.