The involutive quantaloid of completely distributive lattices
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Let L be a complete lattice and let Q(L) be the unital quantale of join-continuous endo-functions of L. We prove the following result: Q(L) is an involutive (that is, non-commutative cyclic -autonomous) quantale if and only if L is a completely distributive lattice. If this is the case, then the dual tensor operation corresponds, via Raney's transforms, to composition in the (dual) quantale of meet-continuous endo-functions of L. Let Latt be the category of sup-lattices and join-continuous functions and let Latt cd be the full subcategory of Latt whose objects are the completely distributive lattices. We argue that (i) Latt cd is itself an involutive quantaloid, and therefore it is the largest full-subcategory of Latt with this property; (ii) Latt cd is closed under the monoidal operations of Latt , and therefore it is star-autonomous. Consequently, if Q(L) is involutive, then it is completely distributive as well.
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