Luigi Santocanale
Current Organizations / Institutes
The organizations/institutes from the scholar's latest article
Previous Organizations / Institutes
The organizations / institutes from the scholar's other articles in the past ten years
Research Subjects
Luigi Santocanale has a total of 22 co-authors. Here are the top co-authors by number of shared publications:
- Silvio Ghilardi
Università degli Studi di Milano
6 shared publications - Cédric de Lacroix
LIS UMR 7020 CNRS Aix-Marseille Université France
5 shared publications - Friedrich Wehrung
Université de Caen Normandie
5 shared publications - Maria João Gouveia
Faculdade de Ciências da Universidade de Lisboa & CAUL, Lisboa, Portugal
5 shared publications Friedrich Wehrung
Université de Caen, France
4 shared publications- Gregory Chichery
LIS UMR 7020 CNRS Aix-Marseille Université France
3 shared publications
Latest Publications
- 12 August 2024
- Published by Springer Nature
- p. 101-118
Complete Congruences of Completely Distributive Lattices
- 12 August 2024
- Published by Springer Nature
- p. 101-118
Complete Congruences of Completely Distributive Lattices
- Open Access
Lifting star-autonomous structures
For a functor $Q$ from a category $C$ to the category $Pos$ of ordered sets and order-preserving functions, we study liftings of various kind of structures from the base category $C$ to the total(or Grothendieck) category $\int Q$. That lifting a monoidal structure corresponds to giving some lax natural transformation making $Q$ almost monoidal, might be part of folklore in category theory.We rely on and generalize the tools supporting this correspondence so to provide exact conditions for lifting symmetric monoidal closed and star-autonomous structures.A corollary of these characterizations is that, if $Q$ factors as a monoidal functor through $SLatt$, the category of complete lattices and sup-preserving functions, then $\int Q$ is always symmetric monoidalclosed. In this case, we also provide a method, based on the double negation nucleus from quantale theory, to turn $\int Q$ into a star-autonomous category.The theory developed, originally motivated from the categories $P-Set$ of Schalk and de Paiva, yields a wide generalization of Hyland and Schalk construction of star-autonomous categories by means of orthogonality structures.Open AccessLifting star-autonomous structures
For a functor $Q$ from a category $C$ to the category $Pos$ of ordered sets and order-preserving functions, we study liftings of various kind of structures from the base category $C$ to the total(or Grothendieck) category $\int Q$. That lifting a monoidal structure corresponds to giving some lax natural transformation making $Q$ almost monoidal, might be part of folklore in category theory.We rely on and generalize the tools supporting this correspondence so to provide exact conditions for lifting symmetric monoidal closed and star-autonomous structures.A corollary of these characterizations is that, if $Q$ factors as a monoidal functor through $SLatt$, the category of complete lattices and sup-preserving functions, then $\int Q$ is always symmetric monoidalclosed. In this case, we also provide a method, based on the double negation nucleus from quantale theory, to turn $\int Q$ into a star-autonomous category.The theory developed, originally motivated from the categories $P-Set$ of Schalk and de Paiva, yields a wide generalization of Hyland and Schalk construction of star-autonomous categories by means of orthogonality structures. - Open Access
Lifting star-autonomous structures
For a functor $Q$ from a category $C$ to the category $Pos$ of ordered sets and order-preserving functions, we study liftings of various kind of structures from the base category $C$ to the total(or Grothendieck) category $\int Q$. That lifting a monoidal structure corresponds to giving some lax natural transformation making $Q$ almost monoidal, might be part of folklore in category theory.We rely on and generalize the tools supporting this correspondence so to provide exact conditions for lifting symmetric monoidal closed and star-autonomous structures.A corollary of these characterizations is that, if $Q$ factors as a monoidal functor through $SLatt$, the category of complete lattices and sup-preserving functions, then $\int Q$ is always symmetric monoidalclosed. In this case, we also provide a method, based on the double negation nucleus from quantale theory, to turn $\int Q$ into a star-autonomous category.The theory developed, originally motivated from the categories $P-Set$ of Schalk and de Paiva, yields a wide generalization of Hyland and Schalk construction of star-autonomous categories by means of orthogonality structures.Open AccessLifting star-autonomous structures
For a functor $Q$ from a category $C$ to the category $Pos$ of ordered sets and order-preserving functions, we study liftings of various kind of structures from the base category $C$ to the total(or Grothendieck) category $\int Q$. That lifting a monoidal structure corresponds to giving some lax natural transformation making $Q$ almost monoidal, might be part of folklore in category theory.We rely on and generalize the tools supporting this correspondence so to provide exact conditions for lifting symmetric monoidal closed and star-autonomous structures.A corollary of these characterizations is that, if $Q$ factors as a monoidal functor through $SLatt$, the category of complete lattices and sup-preserving functions, then $\int Q$ is always symmetric monoidalclosed. In this case, we also provide a method, based on the double negation nucleus from quantale theory, to turn $\int Q$ into a star-autonomous category.The theory developed, originally motivated from the categories $P-Set$ of Schalk and de Paiva, yields a wide generalization of Hyland and Schalk construction of star-autonomous categories by means of orthogonality structures. - 27 December 2022
- Published by Springer Nature in Applied Categorical Structures
- Vol. 31 (1) , 1-24
Unitless Frobenius Quantales
- 27 December 2022
- Published by Springer Nature in Applied Categorical Structures
- Vol. 31 (1) , 1-24
Unitless Frobenius Quantales
- Open Access
Unitless Frobenius quantales
It is often stated that Frobenius quantales are necessarily unital. By taking negation as a primitive operation, we can define Frobenius quantales that may not have a unit. We develop the elementary theory of these structures and show, in particular, how to define nuclei whose quotients are Frobenius quantales. This yields a phase semantics and a representation theorem via phase quantales. Important examples of these structures arise from Raney's notion of tight Galois connection: tight endomaps of a complete lattice always form a Girard quantale which is unital if and only if the lattice is completely distributive. We give a characterisation and an enumeration of tight endomaps of the diamond lattices Mn and exemplify the Frobenius structure on these maps. By means of phase semantics, we exhibit analogous examples built up from trace class operators on an infinite dimensional Hilbert space. Finally, we argue that units cannot be properly added to Frobenius quantales: every possible extention to a unital quantale fails to preserve negations.
Open AccessUnitless Frobenius quantales
It is often stated that Frobenius quantales are necessarily unital. By taking negation as a primitive operation, we can define Frobenius quantales that may not have a unit. We develop the elementary theory of these structures and show, in particular, how to define nuclei whose quotients are Frobenius quantales. This yields a phase semantics and a representation theorem via phase quantales. Important examples of these structures arise from Raney's notion of tight Galois connection: tight endomaps of a complete lattice always form a Girard quantale which is unital if and only if the lattice is completely distributive. We give a characterisation and an enumeration of tight endomaps of the diamond lattices Mn and exemplify the Frobenius structure on these maps. By means of phase semantics, we exhibit analogous examples built up from trace class operators on an infinite dimensional Hilbert space. Finally, we argue that units cannot be properly added to Frobenius quantales: every possible extention to a unital quantale fails to preserve negations.
- Open Access
The Involutive Quantaloid of Completely Distributive Lattices
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 $\star$-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.
Open AccessThe Involutive Quantaloid of Completely Distributive Lattices
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 $\star$-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.
- 1 April 2020
- Published by Springer Nature
- Vol. 12062, 286-301
The Involutive Quantaloid of Completely Distributive Lattices
Let L be a complete lattice and let \({\mathcal {Q}}(L)\) be the unital quantale of join-continuous endo-functions of L. We prove that \({\mathcal {Q}}(L)\) has at most two cyclic elements, and that if it has a non-trivial cyclic element, then L is completely distributive and \({\mathcal {Q}}(L)\) is involutive (that is, non-commutative cyclic \(\star \)-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 \(\mathsf {Latt}_{\bigvee } \) be the category of sup-lattices and join-continuous functions and let \(\mathsf {Latt}_{\bigvee } ^{\mathtt {cd}} \) be the full subcategory of \(\mathsf {Latt}_{\bigvee } \) whose objects are the completely distributive lattices. We argue that \(\mathsf {Latt}_{\bigvee } ^{\mathtt {cd}} \) is itself an involutive quantaloid, thus it is the largest full-subcategory of \(\mathsf {Latt}_{\bigvee }\) with this property. Since \(\mathsf {Latt}_{\bigvee } ^{\mathtt {cd}} \) is closed under the monoidal operations of \(\mathsf {Latt}_{\bigvee } \), we also argue that if \({\mathcal {Q}}(L)\) is involutive, then \({\mathcal {Q}}(L)\) is completely distributive as well; consequently, any lattice embedding into an involutive quantale of the form \({\mathcal {Q}}(L)\) has, as its domain, a distributive lattice.- 1 April 2020
- Published by Springer Nature
- Vol. 12062, 286-301
The Involutive Quantaloid of Completely Distributive Lattices
Let L be a complete lattice and let \({\mathcal {Q}}(L)\) be the unital quantale of join-continuous endo-functions of L. We prove that \({\mathcal {Q}}(L)\) has at most two cyclic elements, and that if it has a non-trivial cyclic element, then L is completely distributive and \({\mathcal {Q}}(L)\) is involutive (that is, non-commutative cyclic \(\star \)-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 \(\mathsf {Latt}_{\bigvee } \) be the category of sup-lattices and join-continuous functions and let \(\mathsf {Latt}_{\bigvee } ^{\mathtt {cd}} \) be the full subcategory of \(\mathsf {Latt}_{\bigvee } \) whose objects are the completely distributive lattices. We argue that \(\mathsf {Latt}_{\bigvee } ^{\mathtt {cd}} \) is itself an involutive quantaloid, thus it is the largest full-subcategory of \(\mathsf {Latt}_{\bigvee }\) with this property. Since \(\mathsf {Latt}_{\bigvee } ^{\mathtt {cd}} \) is closed under the monoidal operations of \(\mathsf {Latt}_{\bigvee } \), we also argue that if \({\mathcal {Q}}(L)\) is involutive, then \({\mathcal {Q}}(L)\) is completely distributive as well; consequently, any lattice embedding into an involutive quantale of the form \({\mathcal {Q}}(L)\) has, as its domain, a distributive lattice.- Open Access
- 15 October 2019
- Published by Centre pour la Communication Scientifique Directe (CCSD) in Logical Methods in Computer Science
- Vol. ume 15, Is
$\aleph_1$ and the modal $\mu$-calculus
For a regular cardinal $\kappa$, a formula of the modal $\mu$-calculus is $\kappa$-continuous in a variable x if, on every model, its interpretation as a unary function of x is monotone and preserves unions of $\kappa$-directed sets. We define the fragment $C_{\aleph_1}(x)$ of the modal $\mu$-calculus and prove that all the formulas in this fragment are $\aleph_1$-continuous. For each formula $\phi(x)$ of the modal $\mu$-calculus, we construct a formula $\psi(x) \in C_{\aleph_1 }(x)$ such that $\phi(x)$ is $\kappa$-continuous, for some $\kappa$, if and only if $\phi(x)$ is equivalent to $\psi(x)$. Consequently, we prove that (i) the problem whether a formula is $\kappa$-continuous for some $\kappa$ is decidable, (ii) up to equivalence, there are only two fragments determined by continuity at some regular cardinal: the fragment $C_{\aleph_0}(x)$ studied by Fontaine and the fragment $C_{\aleph_1}(x)$. We apply our considerations to the problem of characterizing closure ordinals of formulas of the modal $\mu$-calculus. An ordinal $\alpha$ is the closure ordinal of a formula $\phi(x)$ if its interpretation on every model converges to its least fixed-point in at most $\alpha$ steps and if there is a model where the convergence occurs exactly in $\alpha$ steps. We prove that $\omega_1$, the least uncountable ordinal, is such a closure ordinal. Moreover we prove that closure ordinals are closed under ordinal sum. Thus, any formal expression built from 0, 1, $\omega$, $\omega_1$ by using the binary operator symbol + gives rise to a closure ordinal.- 15 October 2019
- Published by Centre pour la Communication Scientifique Directe (CCSD) in Logical Methods in Computer Science
- Vol. ume 15, Is
Open Access$\aleph_1$ and the modal $\mu$-calculus
For a regular cardinal $\kappa$, a formula of the modal $\mu$-calculus is $\kappa$-continuous in a variable x if, on every model, its interpretation as a unary function of x is monotone and preserves unions of $\kappa$-directed sets. We define the fragment $C_{\aleph_1}(x)$ of the modal $\mu$-calculus and prove that all the formulas in this fragment are $\aleph_1$-continuous. For each formula $\phi(x)$ of the modal $\mu$-calculus, we construct a formula $\psi(x) \in C_{\aleph_1 }(x)$ such that $\phi(x)$ is $\kappa$-continuous, for some $\kappa$, if and only if $\phi(x)$ is equivalent to $\psi(x)$. Consequently, we prove that (i) the problem whether a formula is $\kappa$-continuous for some $\kappa$ is decidable, (ii) up to equivalence, there are only two fragments determined by continuity at some regular cardinal: the fragment $C_{\aleph_0}(x)$ studied by Fontaine and the fragment $C_{\aleph_1}(x)$. We apply our considerations to the problem of characterizing closure ordinals of formulas of the modal $\mu$-calculus. An ordinal $\alpha$ is the closure ordinal of a formula $\phi(x)$ if its interpretation on every model converges to its least fixed-point in at most $\alpha$ steps and if there is a model where the convergence occurs exactly in $\alpha$ steps. We prove that $\omega_1$, the least uncountable ordinal, is such a closure ordinal. Moreover we prove that closure ordinals are closed under ordinal sum. Thus, any formal expression built from 0, 1, $\omega$, $\omega_1$ by using the binary operator symbol + gives rise to a closure ordinal.