Hardy type derivations on fields of exponential logarithmic series

We consider the valued field \mathds{K}:=\mathbb{R}((\Gamma)) of formal series (with real coefficients and monomials in a totally ordered multiplicative group $\Gamma>$). We investigate how to endow \mathds{K} with a logarithm $l$, which satisfies some natural properties such as commuting with infinite products of monomials. In the article "Hardy type derivations on generalized series fields", we study derivations on \mathds{K}. Here, we investigate compatibility conditions between the logarithm and the derivation, i.e. when the logarithmic derivative is the derivative of the logarithm. We analyse sufficient conditions on a given derivation to construct a compatible logarithm via integration of logarithmic derivatives. In her monograph "Ordered exponential fields", the first author described the exponential closure \mathds{K}^{\rm{EL}} of (\mathds{K},l). Here we show how to extend such a log-compatible derivation on \mathds{K} to \mathds{K}^{\rm{EL}}.Comment: 25 page

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.Comment: Coalgebraic Methods in Computer Science 2016, Apr 2016, Eindhoven, Netherland

On the Pierce-Birkhoff Conjecture

International audienceThis paper represents a step in our program towards the proof of the Pierce--Birkhoff conjecture. In the nineteen eighties J. Madden proved that the Pierce-Birkhoff conjecture for a ring A$is equivalent to a statement about an arbitrary pair of points$\alpha,\beta\in\sper\ A$and their separating ideal$$; we refer to this statement as the Local Pierce-Birkhoff conjecture at$\alpha,\beta$. In this paper, for each pair$(\alpha,\beta)$with$ht()=\dim A$, we define a natural number, called complexity of$(\alpha,\beta)$. Complexity 0 corresponds to the case when one of the points$\alpha,\beta$is monomial; this case was already settled in all dimensions in a preceding paper. Here we introduce a new conjecture, called the Strong Connectedness conjecture, and prove that the strong connectedness conjecture in dimension n-1 implies the connectedness conjecture in dimension n in the case when$ht()$is less than n-1. We prove the Strong Connectedness conjecture in dimension 2, which gives the Connectedness and the Pierce--Birkhoff conjectures in any dimension in the case when$ht()$less than 2. Finally, we prove the Connectedness (and hence also the Pierce--Birkhoff) conjecture in the case when dimension of A is equal to$ht()=3$, the pair$(\alpha,\beta)$is of complexity 1 and$A$ is excellent with residue field the field of real numbers