arXiv subjects

Maria João Gouveia

Publications and source records attributed to Maria João Gouveia.

3 recordsLinked to original sources

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↗

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↗