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Communication Dans Un Congrès Année : 2020

The Involutive Quantaloid of Completely Distributive Lattices

Résumé

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 sLatt be the category of sup-lattices and join-continuous functions and let cdLatt be the full subcategory of sLatt whose objects are the completely distributive lattices. We argue that (i) cdLatt is itself an involutive quantaloid, and therefore it is the largest full-subcategory of sLatt with this property; (ii) cdLatt is closed under the monoidal operations of sLatt and, consequently, if Q(L) is involutive, then Q(L) is completely distributive as well.
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Dates et versions

hal-02342655 , version 1 (01-11-2019)
hal-02342655 , version 2 (08-11-2019)
hal-02342655 , version 3 (06-04-2020)

Identifiants

Citer

Luigi Santocanale. The Involutive Quantaloid of Completely Distributive Lattices. RAMICS 2020, Uli Fahrenberg; Peter Jipsen; Michael Winter, Apr 2020, Palaiseau, France. pp.286-301. ⟨hal-02342655v3⟩
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