It is argued in (Eklund et al., 2018) that the quantale [L,L] of sup-preserving endomaps of a complete lattice L is a Girard quantale exactly when L is completely distributive. We have argued in (Santocanale, 2020) that this Girard quantale structure arises from the dual quantale of inf-preserving endomaps of L via Raney's transforms and extends to a Girard quantaloid structure on the full subcategory of SLatt (the category of complete lattices and sup-preserving maps) whose objects are the completely distributive lattices. It is the goal of this talk to illustrate further this connection between the quantale structure, Raney's transforms, and complete distributivity. Raney's transforms are indeed mix maps in the isomix category SLatt and most of the theory can be developed relying on naturality of these maps. We complete then the remarks on cyclic elements of [L,L] developed in (Santocanale, 2020) by investigating its dualizing elements. We argue that if [L,L] has the structure a Frobenius quantale, that is, if it has a dualizing element, not necessarily a cyclic one, then L is once more completely distributive. It follows then from a general statement on involutive residuated lattices that there is a bijection between dualizing elements of [L,L] and automorphisms of L. Finally, we also argue that if L is finite and [L,L] is autodual, then L is distributive.

翻译：(Eklund等人,2018年)在(Eklund等人,2018年)中争论道,完整Lattice L 的半透明方程[L,L] 是一个完全分布的Girard 方格。我们在(Santocanale,2020年)中争论道,Girard 方方格结构来自通过Raney的变换保存L的双方方格程的双方程,并延伸到SLatt的整个子类(完整Lattics和sup-reserve 映像的类别),其对象为完全分布的Lattice 方格[L,L] 方格的半透明方格。我们随后完成了SLattrial 和Suprecial 方格中[L] 的双轨元素的评论,而Ltredicial(Santocal,Lal) 则通过对双轨结构进行更深入的研究。如果现在我们是双轨的双轨,那么, 方(L) 方位(L) 的双轨(如果现在是一个双轨) 方系是双轨结构,那么,那么,