Quantale algebras as lattice-valued quantales
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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.
KeywordsQuantale Quantale algebra Q-ordered set Lattice-valued quantale
This work is supported by the National Natural Science Foundation of China (Grant Nos. 11301316, 11531009), the Natural Science Program for Basic Research of Shaanxi Province (Grant No. 2015JM1020) and the Fundamental Research Funds for the Central Universities (Grant Nos. GK201302003, GK201501001). The authors would like to thank the referees and the editors for their valuable comments and suggestions.
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Conflict of interest
The authors declare that they have no conflict of interest.
- Galatos N, Jipsen P, Kowalski T, Ono H (2007) Residuated Lattices: An Algebraic Glimpse at Substructural Logics. In: Studies in Logics and th Foundations of Mathematics, vol 151. Elsevier, AmsterdamGoogle Scholar
- Kelly GM (1982) Basic Concepts of Enriched Category Theory. London Mathematical Society Lecture Notes Series, vol 64. Cambridge University Press, CambridgeGoogle Scholar
- Lai H, Zhang D (2006) Many-valued complete distributivity. arXiv:math/0603590v2
- Mulvey CJ (1986)&. Rend Circ Mat Palermo (2) Suppl 12(2):99–104Google Scholar
- Rodabaugh SE (1999) Powerset operator foundations for poslat fuzzy set theories and topologies. In: Höhle U, Rodabaugh SE (eds) Mathematics of Fuzzy Sets: Logic, Topology, and Measure Theory. The Handbooks of Fuzzy Sets Series, vol 3. Kluwer Academic Publishers, Boston, pp 91–116 (Chapter 2)Google Scholar
- Russo C (2007) Quantale modules, with applications to logic and image processing. PhD thesis, Department of Mathematics and Computer Science, University of Salerno, SalernoGoogle Scholar
- Wagner KR (1994) Solving recursive domain equations with enriched categories. PhD Thesis, School of Computer Science, Carnegie Mellon University, PittsburghGoogle Scholar
- Wang K (2012) Some researches on fuzzy domains and fuzzy quantales. PhD Thesis, Department of Mathematics, Shaanxi Normal University, Xi’an (in Chinese)Google Scholar
- Wang R, Zhao B (2010) Quantale algebra and its algebraic ideal. Fuzzy Syst Math 24:44–49 (in Chinese)Google Scholar
- Wang K, Zhao B (2013) Some properties of the category of fuzzy quantales. J Shaanxi Norm Univ (Nat Sci Ed) 41(3):1–6 (in Chinese)Google Scholar
- Wang K, Zhao B (2016b) A representation theorem for Girard \(Q\)-algebras. J Mult Valued Log Soft Comput (in press)Google Scholar