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Boolean lifting property in quantales

Abstract

In ring theory, the lifting idempotent property (LIP) is related to some important classes of rings: clean rings, exchange rings, local and semilocal rings, Gelfand rings, maximal rings, etc. Inspired by LIP, lifting properties were also defined for other algebraic structures: MV-algebras, BL-algebras, residuated lattices, abelian l-groups, congruence distributive universal algebras, etc. In this paper, we define a lifting property (LP) in commutative coherent integral quantales, structures that are a good abstraction for lattices of ideals, filters and congruences. LP generalizes all the lifting properties existing in the literature. The main tool in the study of LP in a quantale A is the reticulation of A, a bounded distributive lattice whose prime spectrum is homeomorphic to the prime spectrum of A. The principal results of the paper include a characterization theorem for quantales with LP and a characterization theorem for hyperarchimedean quantales.

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Acknowledgements

The authors would like to thank the reviewers for their valuable remarks and comments in order to improve the actual form of the paper.

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Correspondence to Daniela Cheptea.

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Cheptea, D., Georgescu, G. Boolean lifting property in quantales. Soft Comput (2020). https://doi.org/10.1007/s00500-020-04752-8

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Keywords

  • Lifting property
  • Quantale
  • Reticulation
  • Boolean center
  • Normal
  • Semilocal