Luigi Santocanale

Laboratoire d’Informatique et des Systèmes de Marseille

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 r...

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 se...

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 r...

This book constitutes the proceedings of the 19th International Conference on Relational and Algebraic Methods in Computer Science, RAMiCS 2021, which took place in Marseille, France, during November 2-5, 2021. The 29 papers presented in this book were carefully reviewed and selected from 35 submissions. They deal with the development and dissemi...

We consider skew metrics (equivalently, transitive relations that are tournaments, linear orderings) valued in Sugihara semigroups on autodual chains. We prove that, for odd chains and chains without a unit, skew metrics classify certain tree-like structures that we call perfect augmented plane towers. When the chain is finite and has cardinality 2...

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 descent...

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...

It is argued in [5] 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 [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...

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....

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 \(\s...

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 ≥ 0 such that fN +2 = fN , thus the period equals 2. We give a semantic proof of this theorem, using duality techniques and bounded bisimulation ranks. By the same...

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...

It follows from known results in the 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 t...

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 map s from the chain \(\{\,0,1,\ldots ,n\,\}\) to itself. We explicitly describe this monoid struct...

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 ge...

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...

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 $\Sigma$ = {x,y,z,...}, where each letter has a fixed number of occurrences. These lattices are known as multinomial lattices and, when card($\Sigma$) = 2, as lattic...

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 $\Sigma = \{ x, y, z, ... \}$, where each letter has a fixed number of occurrences (these lattices are known as multinomial lattices and, in dimension 2, as lattices...

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 $\...

The natural join and the inner union operations combine relations of a database. Tropashko and Spight [25] 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 rela...

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...

The natural join and the inner union operations combine relations of a database. Tropashko and Spight 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...

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 rela...

The natural join and the inner union operations combine relations of a database. Tropashko and Spight 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...

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...

We can find in the literature many proposals for generalizations of permutohedra. Among those, let us mention the permutohedron on a poset (Pouzet et al. [356]), multinomial lattices (also called lattices of multipermutations, see Bennett and Birkhoff [55], Flath [154], Santocanale [393]), lattices of generalized permutations (Gross [210], Krob et...

Historically, notions or problems related to permutations occur — at least — as early as the Antiquity. For example, among the ancient Greeks, Spartans developed the art of encrypting messages via transposition ciphers as part of their military efforts (cf. Kelly [275]).

George Grätzer’s Lattice Theory: Foundation is his third book on lattice theory (General Lattice Theory, 1978, second edition, 1998). In 2009, Grätzer considered updating the second edition to reflect some exciting and deep developments. He soon realized that to lay the foundation, to survey the contemporary field, to pose research problems, would...

The natural join and the inner union operations combine relations in a database. Tropashko and Spight 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, alternative to the relational algebra, to the theory of...

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 IP...

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] i...

Keywords : relational lattice, natural join, inner union, generalized ultrametric space, OD-graph

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]...

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 IP...

41 pages. A few bugs in the proofs of version 1 are corrected in version 2.

We establish a formal connection between algorithmic correspondence theory and certain dual characterization results for finite lattices, similar to Nation's characterization of a hierarchy of pseudovarieties of finite lattices, progressively generalizing finite distributive lattices. This formal connection is mediated through monotone modal logic....

The varieties of lattices \(\mathcal{D}_n\), n ≥ 0, were introduced in [Nat90] and studied later in [Sem05]. These varieties might be considered as generalizations of the variety of distributive lattices which, as a matter of fact, coincides with \(\mathcal{D}_{0}\). It is well known that least and greatest fixed-points of terms are definable on di...

One of the authors introduced in [16] a calculus of circular proofs for studying the computability arising from the following categorical operations: finite products, finite coproducts, initial algebras, final coalgebras. The calculus presented [16] is cut-free; even if sound and complete for provability, it lacked an important property for the sem...

For a closure space (P,f) with f(\emptyset)=\emptyset, the closures of open subsets of P, called the regular closed subsets, form an ortholattice Reg(P,f), extending the poset Clop(P,f) of all clopen subsets. If (P,f) is a finite convex geometry, then Reg(P,f) is pseudocomplemented. The Dedekind-MacNeille completion of the poset of regions of any c...

For a given transitive binary relation e on a set E, the transitive closures of open (i.e., co-transitive in e) sets, called the regular closed subsets, form an ortholattice Reg(e), the extended permutohedron on e. This construction, which contains the poset Clop(e) of all clopen sets, is a common generalization of known notions such as the general...

Gr\"atzer asked in 1971 for a characterization of sublattices of Tamari lattices (associahedra). A natural candidate was coined by McKenzie in 1972 with the notion of a bounded homomorphic image of a free lattice---in short, bounded lattice. Urquhart proved in 1978 that every associahedron is bounded (thus so are its sublattices). Geyer conjectured...

A lattice L is spatial if every element of L is a join of completely join-irreducible elements of L (points), and strongly spatial if it is spatial and the minimal coverings of completely join-irreducible elements are well-behaved. Herrmann, Pickering, and Roddy proved in 1994 that every modular lattice can be embedded, within its variety, into an...

Parity games are combinatorial representations of closed Boolean μ-terms. By adding to them draw positions, they have been organized by Arnold and Santocanale (2005, 2007) [3,27] into a μ-calculus (Arnold, 2001 [2]) whose standard interpretation is over the class of all complete lattices. As done by Berwanger et al. (2002, 2005) [8,9] for the propo...

We reconstruct the syntax and semantics of monotone modal logic, in the style of Moss’s coalgebraic logic. To that aim, we replace the box and diamond with a modality ∇ which takes a finite collection of finite sets of formulas as its argument. The semantics of this modality in monotone neighborhood models is defined in terms of a version of relati...

A presentation is a triple $\leangle X,\leq,M\rangle$ with $\langle X,\leq\rangle$ a finite poset and $M : X \rTo \P(\P(X))$ -- these data being subject to additional constraints. Given a presentation we can define closed subsets of $X$, whence a finite lattice. Given a finite lattice $L$, we can define its presentation: $X$ is the set of join-irre...

This paper presents a feasible decision procedure for the equality of parallel arrows in the initial category with finite products and coproducts. The algorithm, in particular, handles the “additive units” and demonstrates that the complications introduced by the presence of these units can be managed in an efficient manner. This problem is directl...

We address the problem of finding nice labellings for event structures of degree 3. We develop a minimum theory by which we prove that the labelling number of an event structure of degree 3 is bounded by a linear function of the height. The main theorem we present in this paper states that event structures of degree 3 whose causality order is a tre...

Motivated by the nice labelling problem for event structures, we study the topological properties of the associated graphs. For each n⩾0, we exhibit a graph Gn that cannot occur on an antichain as a subgraph of the graph of an event structure of degree n. The clique complexes of the graphs Gn are disks (n even) and spheres (n odd) in increasing dim...

Given a set Γ of modal formulas of the form γ(x, p), where x occurs positively in γ, the language \(\mathcal{L}_\sharp({\it \Gamma})\) is obtained by adding to the language of polymodal logic K connectives \(\sharp_\gamma\), γε Γ. Each term \(\sharp_\gamma\) is meant to be interpreted as the parametrized least fixed point of the functional interpre...

This paper presents a feasible decision procedure for the equality of parallel arrows in the initial category with finite products and coproducts. The algorithm, in particular, handles the “additive units” and demonstrates that the complications introduced by the presence of these units can be managed in an efficient manner. This problem is direct...

Dans cet ouvrage nous allons résumer nos activités de recherche depuis l'obtention du titre de docteur à l'Université du Québec à Montréal. Ces recherches ont eu lieu auprès de et ont été possibles grâce à de nombreuses institutions que nous remercions : le BRICS à l'Université de Aarhus, le PIMS et le Département d'Informatique de l'Université de...

A μ-algebra is a model of a first-order theory that is an extension of the theory of bounded lattices, that comes with pairs of terms (f,μx.f) where μx.f is axiomatized as the least prefixed point of f, whose axioms are equations or equational implications.Standard μ-algebras are complete meaning that their lattice reduct is a complete lattice. We...

Entanglement is a complexity measure of directed graphs that origins in fixed point theory. This measure has shown its use in designing efficient algorithms to verify logical propertiesof transition systems. We are interested in the problem of deciding whether a graph has entanglement at most k. As this measure is defined by means of games, game th...

Aiming to understand equivalence relations that model concurrent computation, we investigate congruences of multinomial lattices \(\mathcal{L}(v)\) introduced by Bennett and Birkhoff (Algebra Univers. 32(1):115–144, 1994). Our investigation gives rise to an explicit description of the join dependency relation between two join irreducible elements a...

Parity games are combinatorial representations of closed Boolean mu-terms. By adding to them draw positions, they have been organized by Arnold and one of the authors into a mu-calculus. As done by Berwanger et al. for the propositional modal mu-calculus, it is possible to classify parity games into levels of a hierarchy according to the number of...

For L a finite lattice, let \({\mathbb {C}(L) \subseteq L^2}\) denote the set of pairs γ = (γ 0, γ 1) such that \({\gamma_0 \prec \gamma_1}\) and order it as followsγ ≤ δ iff γ 0 ≤ δ 0, \({\gamma_{1} \nleq \delta_0,}\) and γ 1 ≤ δ 1. Let \({\mathbb {C}(L, \gamma)}\) denote the connected component of γ in this poset. Our main result states that, for...

We address the problem of finding nice labellings for event structures of degree 3. We develop a minimum theory by which we prove that the index of an event structure of degree 3 is bounded by a linear function of the height. The main theorem of the paper states that event structures of degree 3 whose causality order is a tree have a nice labelling...

We study the congruence lattices of the multinomial lattices L(v) introduced by Bennett and Birkhoff. Our main motivation is to investigate Parikh equivalence relations that model concurrent computation. We accomplish this goal by providing an explicit description of the join dependency relation between two join irreducible elements and of its refl...

A $\mu$-algebra is a model of a first order theory that is an extension of the theory of bounded lattices, that comes with pairs of terms $(f,\mu_{x}.f)$ where $\mu_{x}.f$ is axiomatized as the least prefixed point of $f$, whose axioms are equations or equational implications. Standard $\mu$-algebras are complete meaning that their lattice reduct i...

We define the class of algebraic models of μ-calculi and study whether every such model can be embedded into a model which is a complete lattice. We show that this is false in the general case and focus then on free modal μ-algebras, i.e. Lindenbaum algebras of the propositional modal μ-calculus. We prove the following fact: the MacNeille-Dedekind...

Abstract A classical result by Rabin states that if a set of trees and its complement,are both Büchi definable in the monadic second order logic, then these sets are weakly definable. In the language of -calculi, this theorem asserts the equality between the complexity classes 2 ∩2 and Comp(1,1) of the fixed-point alternation-depth hierarchy of the...

We introduce a new method (derived from model theoretic general combination procedures in automated deduction) for proving fusion decidability in modal systems. We apply it to show fusion decidability in case not only the boolean connectives, but also a universal modality and nominals are shared symbols.

We prove that every finitary polynomial endofunctor of a category has a final coalgebra, provided that is locally Cartesian closed, it has finite coproducts and is an extensive category, it has a natural number object.

Every parity game is a combinatorial representation of a closed Boolean μ-term. When interpreted in a distributive lattice every Boolean μ-term is equivalent to a fixed-point free term. The alternationdepth hierarchy is therefore trivial in this case. This is not the case for non distributive lattices, as the second author has shown that the altern...

We prove that every nitary polynomial endofunctor of a category C has a nal coalgebra, provided that C is locally Cartesian closed, it has nite coproducts and is an extensive category, it has a natural number object. 1

We survey on the ongoing research that relates the combinatorics of parity games to the algebra of categories with nite products, nite coproducts, initial algebras and nal coalgebras of de nable functors, i.e. -bicomplete categories.

We investigate the reasons for which the existence of certain right adjoints implies the existence of some nal coalgebras, and vice-versa. In particular we prove and discuss the following theorem which has been partially available in the literature: let F a G be a pair of adjoint functors, and suppose that an initial algebra F (X) of the functor H(...

We survey on the ongoing research that relates the combinatorics of parity games to the algebra of categories with finite products, finite coproducts, initial algebras and final coalgebras of definable functors, i.e. µ-bicomplete categories. We argue that parity games with a given starting position play the role of terms for the theory of µ-bicompl...

for a given player. The two dierent meanings of parity games, the algebraic one and the combinatorial one, are then shown to coincide. By means of this result we support the claim that the algebra of parity games is the one of -bicomplete categories and that the combinatorics of -bicomplete categories is the one of parity games. 1 -Bicomplete Categ...

The alternation hierarchy problem asks whether every mu-term, that is a term built up using also a least fixed point constructor as well as a greatest fixed point constructor, is equivalent to a mu-term where the number of nested fixed point of a different type is bounded by a fixed number. In this paper we give a proof that the alternation hierarc...

we restrict our sematic domain to complete lattices. . The alternation hierarchy for the theory of -lattices is strict [12]. The characterisation of free -lattices is achieved by describing a class (X) of games with a payo# function with values in X ; games in this class are strictly related to parity games [4] and to those games which have been st...

For an arbitrary category, we consider the least class of functors containing 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 provid...

A μ-lattice is a lattice with the property that every unary polynomial has both a least and a greatest fix-point. In this paper we define the quasivariety of μ-lattices and, for a given partially ordered set P, we construct a μ-lattice whose elements are equivalence classes of games in a preordered class . We prove that the μ-lattice is free over t...

A mu-lattice is a lattice with the property that every unary polynomial has both a least and a greatest fix-point. In this paper we define the quasivariety of mu-lattices and, for a given partially ordered set P, we construct a mu-lattice JP whose elements are equivalence classes of games in a preordered class J (P). We prove that the mu-lattice J...

The alternation hierarchy problem asks whether every -term #, that is, a term built up also using a least fixed point constructor as well as a greatest fixed point constructor, is equivalent to a -term where the number of nested fixed points of a di#erent type is bounded by a constant independent of #.

We present a calculus of circular proofs": the graph underlying a proof is not a nite tree but instead it is allowed to contain a certain amount of cycles. The main challenge in developing a theory for the calculus is to de ne the semantics of proofs, since the usual method by induction on the structure is not available. We solve this problem by as...

We investigate the reasons for which the existence of certain right adjoints implies the existence of some final coalgebras, and vice-versa. In particular we prove and discuss the following theorem which has been partially available in the literature: let F a G be a pair of adjoint functors, and suppose that an initial algebra b F(X) of the functor...

Axioms ruling linear negation have been investigated in the context of the complete semantics for distributive intuitionistic linear logic. Among these are the condition of being a dualizing element and the one of being a cyclic element. The motivation for analyzing other syntactic constraints comes from the observation that groupoids are models fo...

We propose a method to axiomatize by equations the least prefixed point of an order preserving function. We discuss its domain of application and show that the Boolean modal μ-calculus has a complete equational axiomatization. The method relies on the existence of a “closed structure” and its relationship to the equational axiomatization of Action...

The variable hierarchy problem asks whether every μ-term t is equivalent to a μ-term t′ where the number of fixed-point variables in t′ is bounded by a constant. In this paper we prove that the variable hierarchy of the lattice μ -calculus – whose standard interpretation is over the class of all complete lattices – is infinite, meaning that such a...

We address the problem of finding nice labellings for event structures of degree 3. We develop a minimum theory by which we prove that the labelling number of an event structure of degree 3 is bounded by a linear function of the height. The main theorem we present in this paper states that event structures of degree 3 whose causality order is a tre...

An event structures is a mathematical model of a concurrent process. It consists of a set of local events ordered by a causality relation and separated by a conflict relation. A global state, or configuration, is an order ideal whose elements are pairwise not in conflict. Configurations, ordered by subset inclusion, form a poset whose Hasse diagram...