Soft Computing

, Volume 21, Issue 10, pp 2561–2574

Quantale algebras as lattice-valued quantales


DOI: 10.1007/s00500-016-2147-5

Cite this article as:
Zhao, B., Wu, S. & Wang, K. Soft Comput (2017) 21: 2561. doi:10.1007/s00500-016-2147-5


In this paper, we present an investigation of quantale algebras (Q-algebras for short) as lattice-valued quantales (Q-quantales for short). First, we prove that the set of all fuzzy ideals of a commutative ring with appropriate operations is a [0, 1]-quantale. Furthermore, we discuss some properties of localic nuclei on Q-algebras, and show that the category of Q-algebras with the quantale structures being frames is a full reflective subcategory of the category of Q-algebras. From this result, we can conclude that the category of L-frames is a full reflective subcategory of the category of L-quantales, where L is a frame. Finally, we build and characterize the Q-quantale completions of a Q-ordered semigroup.


Quantale Quantale algebra Q-ordered set Lattice-valued quantale 

Funding information

Funder NameGrant NumberFunding Note
the National Natural Science Foundation of China
  • 11301316,11531009

Copyright information

© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  1. 1.College of Mathematics and Information ScienceShaanxi Normal UniversityXi’anPeople’s Republic of China

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