arXiv subjects

Luigi Santocanale

Publications and source records attributed to Luigi Santocanale.

At least 19 recordsLinked to original sources

arXiv2023

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.

cs.LO↗

arXiv2023

Bijective proofs for Eulerian numbers of types B and D

Let $\Bigl\langle\matrix{n\cr k}\Bigr\rangle$, $\Bigl\langle\matrix{B_n\cr k}\Bigr\rangle$, and $\Bigl\langle\matrix{D_n\cr k}\Bigr\rangle$ be the Eulerian numbers in the types A, B, and D, respectively -- that is, the number of permutations of n elements with $k$ descents, the number of signed permutations (of $n$ elements) with $k$ type B descents, the number of even signed permutations (of $n$ elements) with $k$ type D descents. Let $S_n(t) = \sum_{k = 0}^{n-1} \Bigl\langle\matrix{n\cr k}\Bigr\rangle t^k$, $B_n(t) = \sum_{k = 0}^n \Bigl\langle\matrix{B_n\cr k}\Bigr\rangle t^k$, and $D_n(t) = \sum_{k = 0}^n \Bigl\langle\matrix{D_n\cr k}\Bigr\rangle t^k$. We give bijective proofs of the identity $$B_n(t^2) = (1 + t)^{n+1}S_n(t) - 2^n tS_n(t^2)$$ and of Stembridge's identity $$D_n(t) = B_n(t) - n2^{n-1}tS_{n-1}(t).$$ These bijective proofs rely on a representation of signed permutations as paths. Using this representation we also establish a bijective correspondence between even signed permutations and pairs $(w, E)$ with $([n], E)$ a threshold graph and $w$ a degree ordering of $([n], E)$, which we use to obtain bijective proofs of enumerative results for threshold graphs.

cs.LO↗

arXiv2022

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.

cs.LO↗

arXiv2022

Frobenius structures in star-autonomous categories

It is known that the quantale of sup-preserving maps from a complete lattice to itself is a Frobenius quantale if and only if the lattice is completely distributive. Since completely distributive lattices are the nuclear objects in the autonomous category of complete lattices and sup-preserving maps, we study the above statement in a categorical setting. We introduce the notion of Frobenius structure in an arbitrary autonomous category, generalizing that of Frobenius quantale. We prove that the monoid of endomorphisms of a nuclear object has a Frobenius structure. If the environment category is star-autonomous and has epi-mono factorizations, a variant of this theorem allows to develop an abstract phase semantics and to generalise the previous statement. Conversely, we argue that, in a star-autonomous category where the monoidal unit is a dualizing object, if the monoid of endomorphisms of an object has a Frobenius structure and the monoidal unit embeds into this object as a retract, then the object is nuclear.

cs.LO↗

arXiv2021

Dualizing sup-preserving endomaps of a complete lattice

It is argued in (Eklund et al., 2018) that the quantale [L,L] of sup-preserving endomaps of a complete lattice L is a Girard quantale exactly when L is completely distributive. We have argued in (Santocanale, 2020) 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. We complete then the remarks on cyclic elements of [L,L] developed in (Santocanale, 2020) 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. It follows then from a general statement on involutive residuated lattices that there is a bijection between dualizing elements of [L,L] and automorphisms of L. Finally, we also argue that if L is finite and [L,L] is autodual, then L is distributive.

cs.LO↗

arXiv2020

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.

cs.LO↗

arXiv2019

$\aleph_1$ and the modal $μ$-calculus

For a regular cardinal $κ$, a formula of the modal $μ$-calculus is $κ$-continuous in a variable x if, on every model, its interpretation as a unary function of x is monotone and preserves unions of $κ$-directed sets. We define the fragment $C_{\aleph_1}(x)$ of the modal $μ$-calculus and prove that all the formulas in this fragment are $\aleph_1$-continuous. For each formula $ϕ(x)$ of the modal $μ$-calculus, we construct a formula $ψ(x) \in C_{\aleph_1 }(x)$ such that $ϕ(x)$ is $κ$-continuous, for some $κ$, if and only if $ϕ(x)$ is equivalent to $ψ(x)$. Consequently, we prove that (i) the problem whether a formula is $κ$-continuous for some $κ$ 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 $μ$-calculus. An ordinal $α$ is the closure ordinal of a formula $ϕ(x)$ if its interpretation on every model converges to its least fixed-point in at most $α$ steps and if there is a model where the convergence occurs exactly in $α$ steps. We prove that $ω_1$, 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, $ω$, $ω_1$ by using the binary operator symbol + gives rise to a closure ordinal.

cs.LO↗

arXiv2019

On discrete idempotent paths

The set of discrete lattice paths from (0, 0) to (n, n) with North and East steps (i.e. words w $\in$ { x, y } * such that |w| x = |w| y = n) has a canonical monoid structure inherited from the bijection with the set of join-continuous maps from the chain { 0, 1,. .. , n } to itself. We explicitly describe this monoid structure and, relying on a general characterization of idempotent join-continuous maps from a complete lattice to itself, we characterize idempotent paths as upper zigzag paths. We argue that these paths are counted by the odd Fibonacci numbers. Our method yields a geometric/combinatorial proof of counting results, due to Howie and to Laradji and Umar, for idempotents in monoids of monotone endomaps on finite chains.

math.LO↗

arXiv2019

Free Heyting Algebra Endomorphisms: Ruitenburg's Theorem and Beyond

Ruitenburg's Theorem says that every endomorphism f of a finitely generated free Heyting algebra is ultimately periodic if f fixes all the generators but one. More precisely, there is N $\ge$ 0 such that f N +2 = f N , thus the period equals 2. We give a semantic proof of this theorem, using duality techniques and bounded bisimulation ranks. By the same techniques, we tackle investigation of arbitrary endomorphisms between free algebras. We show that they are not, in general, ultimately periodic. Yet, when they are (e.g. in the case of locally finite subvarieties), the period can be explicitly bounded as function of the cardinality of the set of generators.

math.LO↗

arXiv2018

The continuous weak order

The set of permutations on a finite set can be given the lattice structure known as the weak Bruhat order. This lattice structure is generalized to the set of words on a fixed alphabet $Σ$ = {x,y,z,...}, where each letter has a fixed number of occurrences. These lattices are known as multinomial lattices and, when card($Σ$) = 2, as lattices of lattice paths. By interpreting the letters x, y, z, . . . as axes, these words can be interpreted as discrete increasing paths on a grid of a d-dimensional cube, with d = card($Σ$).We show how to extend this ordering to images of continuous monotone functions from the unit interval to a d-dimensional cube and prove that this ordering is a lattice, denoted by L(I^d). This construction relies on a few algebraic properties of the quantale of join-continuous functions from the unit interval of the reals to itself: it is cyclic $\star$-autonomous and it satisfies the mix rule.We investigate structural properties of these lattices, which are self-dual and not distributive. We characterize join-irreducible elements and show that these lattices are generated under infinite joins from their join-irreducible elements, they have no completely join-irreducible elements nor compact elements. We study then embeddings of the d-dimensional multinomial lattices into L(I^d). We show that these embeddings arise functorially from subdivisions of the unit interval and observe that L(I^d) is the Dedekind-MacNeille completion of the colimit of these embeddings. Yet, if we restrict to embeddings that take rational values and if d > 2, then every element of L(I^d) is only a join of meets of elements from the colimit of these embeddings.

math.CO↗

arXiv2018

Mix $\star$-autonomous quantales and the continuous weak order

The set of permutations on a finite set can be given a lattice structure (known as the weak Bruhat order). The lattice structure is generalized to the set of words on a fixed alphabet $Σ= \{ x, y, z, ... \}$, where each letter has a fixed number of occurrences (these lattices are known as multinomial lattices and, in dimension 2, as lattices of lattice paths). By interpreting the letters $x, y, z, ...$ as axes, these words can be interpreted as discrete increasing paths on a grid of a $d$-dimensional cube, where $d = {\rm card}(Σ)$. We show in this paper how to extend this order to images of continuous monotone paths from the unit interval to a $d$-dimensional cube. The key tool used to realize this construction is the quantale $\mathsf{L}_{\vee}(\mathbb{I})$ of join-continuous functions from the unit interval to itself; the construction relies on a few algebraic properties of this quantale: it is $\star$-autonomous and it satisfies the mix rule. We begin developing a structural theory of these lattices by characterizing join-irreducible elements, and by proving these lattices are generated from their join-irreducible elements under infinite joins.

math.LO↗

arXiv2018

Ruitenburg's Theorem via Duality and Bounded Bisimulations

For a given intuitionistic propositional formula A and a propositional variable x occurring in it, define the infinite sequence of formulae { A \_i | i$\ge$1} by letting A\_1 be A and A\_{i+1} be A(A\_i/x). Ruitenburg's Theorem [8] says that the sequence { A \_i } (modulo logical equivalence) is ultimately periodic with period 2, i.e. there is N $\ge$ 0 such that A N+2 $\leftrightarrow$ A N is provable in intuitionistic propositional calculus. We give a semantic proof of this theorem, using duality techniques and bounded bisimulations ranks.

math.LO↗

arXiv2018

Fixed-point elimination in the Intuitionistic Propositional Calculus (extended version)

It is a consequence of existing literature that least and greatest fixed-points of monotone polynomials on Heyting algebras-that is, the alge- braic models of the Intuitionistic Propositional Calculus-always exist, even when these algebras are not complete as lattices. The reason is that these extremal fixed-points are definable by formulas of the IPC. Consequently, the $μ$-calculus based on intuitionistic logic is trivial, every $μ$-formula being equiv- alent to a fixed-point free formula. We give in this paper an axiomatization of least and greatest fixed-points of formulas, and an algorithm to compute a fixed-point free formula equivalent to a given $μ$-formula. The axiomatization of the greatest fixed-point is simple. The axiomatization of the least fixed- point is more complex, in particular every monotone formula converges to its least fixed-point by Kleene's iteration in a finite number of steps, but there is no uniform upper bound on the number of iterations. We extract, out of the algorithm, upper bounds for such n, depending on the size of the formula. For some formulas, we show that these upper bounds are polynomial and optimal.

math.LO↗

arXiv2017

The equational theory of the natural join and inner union is decidable

The natural join and the inner union operations combine relations of a database. Tropashko and Spight [24] realized that these two operations are the meet and join operations in a class of lattices, known by now as the relational lattices. They proposed then lattice theory as an algebraic approach to the theory of databases, alternative to the relational algebra. Previous works [17, 22] proved that the quasiequational theory of these lattices-that is, the set of definite Horn sentences valid in all the relational lattices-is undecidable, even when the signature is restricted to the pure lattice signature. We prove here that the equational theory of relational lattices is decidable. That, is we provide an algorithm to decide if two lattice theoretic terms t, s are made equal under all intepretations in some relational lattice. We achieve this goal by showing that if an inclusion t $\le$ s fails in any of these lattices, then it fails in a relational lattice whose size is bound by a triple exponential function of the sizes of t and s.

cs.LO↗

arXiv2017

The quasiequational theory of relational lattices, in the pure lattice signature (embeddability into relational lattices is undecidable)

The natural join and the inner union operations combine relations of a database. Tropashko and Spight realized that these two operations are themeet and join operations in a class of lattices, known by now as the relational lattices. They proposed then lattice theory as an algebraic approach to thetheory of databases alternative to the relational algebra. Litak et al. proposed an axiomatization of relational lattices over the signature that extends thepure lattice signature with a constant and argued that the quasiequational theory of relational lattices over this extended signature is undecidable.We prove in this paper that embeddability is undecidable for relational lattices. More precisely, it is undecidable whether a finite subdirectly-irreduciblelattice can be embedded into a relational lattice. Our proof is a reduction from the coverability problem of a multimodal frame by a universal product frameand, indirectly, from the representability problem for relation algebras. As corollaries we obtain the following results: the quasiequational theoryof relational lattices over the pure lattice signature is undecidable and has no finite base; there is a quasiequation over the pure lattice signature which holds in all the finite relational lattices but fails in an infinite relational lattice.

cs.LO↗

arXiv2016

$μ$-Bicomplete Categories and Parity Games

For an arbitrary category, we consider the least class of functors con- taining the projections and closed under finite products, finite coproducts, parameterized initial algebras and parameterized final coalgebras, i.e. the class of functors that are definable by $μ$-terms. We call the category $μ$-bicomplete if every $μ$-term defines a functor. We provide concrete ex- amples of such categories and explicitly characterize this class of functors for the category of sets and functions. This goal is achieved through par- ity games: we associate to each game an algebraic expression and turn the game into a term of a categorical theory. We show that $μ$-terms and parity games are equivalent, meaning that they define the same property of being $μ$-bicomplete. Finally, the interpretation of a parity game in the category of sets is shown to be the set of deterministic winning strategies for a chosen player.

cs.LO↗

arXiv2016

Relational lattices via duality

The natural join and the inner union combine in different ways tables of a relational database. Tropashko [18] observed that these two operations are the meet and join in a class of lattices-called the relational lattices- and proposed lattice theory as an alternative algebraic approach to databases. Aiming at query optimization, Litak et al. [12] initiated the study of the equational theory of these lattices. We carry on with this project, making use of the duality theory developed in [16]. The contributions of this paper are as follows. Let A be a set of column's names and D be a set of cell values; we characterize the dual space of the relational lattice R(D, A) by means of a generalized ultrametric space, whose elements are the functions from A to D, with the P (A)-valued distance being the Hamming one but lifted to subsets of A. We use the dual space to present an equational axiomatization of these lattices that reflects the combinatorial properties of these generalized ultrametric spaces: symmetry and pairwise completeness. Finally, we argue that these equations correspond to combinatorial properties of the dual spaces of lattices, in a technical sense analogous of correspondence theory in modal logic. In particular, this leads to an exact characterization of the finite lattices satisfying these equations.

cs.LO↗

arXiv2016

Fixed-point elimination in the intuitionistic propositional calculus

It is a consequence of existing literature that least and greatest fixed-points of monotone polynomials on Heyting algebras-that is, the algebraic models of the Intuitionistic Propositional Calculus-always exist, even when these algebras are not complete as lattices. The reason is that these extremal fixed-points are definable by formulas of the IPC. Consequently, the $μ$-calculus based on intuitionistic logic is trivial, every $μ$-formula being equivalent to a fixed-point free formula. We give in this paper an axiomatization of least and greatest fixed-points of formulas, and an algorithm to compute a fixed-point free formula equivalent to a given $μ$-formula. The axiomatization of the greatest fixed-point is simple. The axiomatization of the least fixed-point is more complex, in particular every monotone formula converges to its least fixed-point by Kleene's iteration in a finite number of steps, but there is no uniform upper bound on the number of iterations. We extract, out of the algorithm, upper bounds for such n, depending on the size of the formula. For some formulas, we show that these upper bounds are polynomial and optimal.

cs.LO↗