Fundamentals of Quantales

Eklund, Patrik; García, Javier Gutiérrez; Höhle, Ulrich; Kortelainen, Jari

doi:10.1007/978-3-319-78948-4_2

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“…In this way we also obtain a refinement of one side of the equivalence stated in Proposition 2.6.18 of [ 4 ], where we do not need to refer to the cyclic dualizing element.…”

Section: Cyclic Elements Ofmentioning

confidence: 99%

“…Let us use for the unital quantale of join-continuous endo-functions of a complete lattice L . Recalling that involutive quantale structures on a given quantale are determined by the cyclic dualizing elements and that complete chains are completely distributive, the following statement from the monograph [ 4 ] shows that is involutive for each complete chain C .…”

Section: Introductionmentioning

confidence: 99%

“…2.6.18 in [ 4 ] . Let L be a complete lattice and let be the join-continuous self-mapping on L defined by , for .…”

Section: Introductionmentioning

confidence: 99%

“…If is cyclic, then either c is the top element of or . Moreover, if is cyclic and not equal to the top element of , then L is completely distributive (and therefore is involutive, as from Proposition 2.6.18 of [ 4 ]).…”

Section: Introductionmentioning

confidence: 99%

“…It is known that the full subcategory of the category of complete lattices and join-continuous functions whose objects are the completely distributive lattices, the involutive quantaloid of completely distributive lattice, is closed under the monoidal operations inherited from the super category, see e.g. [ 4 , 7 , 20 ]. In particular, this quantaloid is itself a -autonomous category.…”

Section: Introductionmentioning

confidence: 99%

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The Involutive Quantaloid of Completely Distributive Lattices

Santocanale¹

2020

Lecture Notes in Computer Science

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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.

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“…In this way we also obtain a refinement of one side of the equivalence stated in Proposition 2.6.18 of [ 4 ], where we do not need to refer to the cyclic dualizing element.…”

Section: Cyclic Elements Ofmentioning

confidence: 99%