Algebra universalis

, 80:52 | Cite as

On the quantale of quantic nuclei

  • Shengwei HanEmail author


It is well known that the set \(\textit{NL}\) of all nuclei on a frame L is again a frame with the pointwise order. As a non-commutative generalization of frames, quantales were introduced by Mulvey in 1986. Niefield and Rosenthal attempted to define a binary operation  & on the set \(\textit{NQ}\) of all quantic nuclei of a quantale Q such that (\(\textit{NQ}\), &) is again a quantale. However, Sun showed that the way of Niefield and Rosenthal failed for a general quantale. Girard quantales are a special class of quantales, which play an important role in the study of linear intuitionistic logic, quantitative domain, lattice-valued topology and enriched category. In this note, we give an example to indicate that \(\textit{NQ}\), together with  & defined by Niefield and Rosenthal, is in general not a quantale even for a Girard quantale Q. However, with respect to  & on a Girard quantale Q, we redefine a binary operation \(\odot \) on \(\textit{NQ}\) such that \((\textit{NQ}^{op}, \odot )\) is a quantale.


Quantale Girard quantale Ideal Quantic nucleus 

Mathematics Subject Classification

06F07 06A06 54A10 



I would like to thank the referees for some of their comments and suggestions for the improvement of this paper.


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© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Department of MathematicsShaanxi Normal UniversityXi’anChina

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