Abstract
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.
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I would like to thank the referees for some of their comments and suggestions for the improvement of this paper.
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This work is supported by the National Natural Science Foundation of China (Grant nos: 11971286, 11531009, 11871320).
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Han, S. On the quantale of quantic nuclei. Algebra Univers. 80, 52 (2019). https://doi.org/10.1007/s00012-019-0627-z
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DOI: https://doi.org/10.1007/s00012-019-0627-z