The Involutive Quantaloid of Completely Distributive Lattices

Abstract

Let L be a complete lattice and let be the unital quantale of join-continuous endo-functions of L . We prove that has at most two cyclic elements, and that if it has a non-trivial cyclic element, then L is completely distributive and is involutive (that is, non-commutative cyclic -autonomous). 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 be the category of sup-lattices and join-continuous functions and let be the full subcategory of whose objects are the completely distributive lattices. We argue that is itself an involutive quantaloid, thus it is the largest full-subcategory of with this property. Since is closed under the monoidal operations of , we also argue that if is involutive, then is completely distributive as well; consequently, any lattice embedding into an involutive quantale of the form has, as its domain, a distributive lattice.

Time Warps, from Algebra to Algorithms

“…))), and, by ( 4), also (15), and f extends ⌊t⌋ δ , we have 6) and (18). So, using the previous cases, (4), and the fact that f extends ⌊t⌋ δ , we get f r It remains to show that the extension is strong.…”

“…with a least element ⊥ that maps all p ∈ ω + to 0, and a greatest element ⊤ that maps all p ∈ ω + \{0} to ω. Note that the operation • is a double quasi-operator on this lattice in the sense of [7,8], and that the structure W , ∧, ∨, •, id belongs to the family of unital quantales of sup-preserving functions on a complete lattice studied in [18].…”

Dualizing sup-preserving endomaps of a complete lattice

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∨ of sup-preserving endomaps of a complete lattice L is a Girard quantale exactly when L is completely distributive. We have argued in [16] 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.

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“…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 [16] 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.…”

“…∨ of sup-preserving endomaps of a complete lattice L is a Girard quantale exactly when L is completely distributive. We have argued in [16] 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.…”