On the quantale of quantic nuclei

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.