Quantale algebras as lattice-valued quantales
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.
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