arXiv subjects

Paul Poncet

Publications and source records attributed to Paul Poncet.

At least 19 recordsLinked to original sources

arXiv2026

Polynomials over idempotent semifields

We study univariate polynomials with coefficients in an idempotent semifield and their factorization. We do not assume the idempotent semifield under consideration to be totally ordered, in contrast with most of the existing work on this topic. We notably determine when a polynomial splits into linear factors, and when its associated polynomial function does so. These results lead us to characterize algebraically closed idempotent semifields -- those in which every polynomial function splits. We prove in particular that every complete idempotent semifield is algebraically closed. We also relate algebraic closedness to the properties of preradicability and radicability and to the existence of solutions to polynomial equations or inequalities.

math.AC↗

arXiv2025

Order-generation in posets and convolution of closure operators

Motivated by the Hofmann--Lawson theorem, which states that every continuous lattice is inf-generated by its irreducible elements, we explore how to represent posets by extreme points with respect to a closure operator. For this purpose, we introduce the convolution product of closure operators, and prove that the Krein--Milman property can be transferred from one collection of subsets to another by convolution. This result underpins two new representation theorems of topological flavor, which generalize existing ones, even in posets lacking lattice or semilattice structures. We also prove a third representation theorem: given a poset equipped with a closure operator $\mathfrak{c}$ with adequate properties, we show that the set of kit points, defined as an extension of compact points, has the Krein--Milman property with respect to the convolution product of $\mathfrak{c}$ with the dual Alexandrov operator $\uparrow\!\! \cdot$; moreover, every kit point is sup-generated by a unique antichain of compact points, finite if $\mathfrak{c}$ is finitary.

math.CO↗

arXiv2024

Partial metrics and normed inverse semigroups

Relying on the notions of submodular function and partial metric, we introduce normed inverse semigroups as a generalization of normed groups and sup-semilattices equipped with an upper valuation. We define the property of skew-convexity for a metric on an inverse semigroup, and prove that every norm on a Clifford semigroup gives rise to a right-subinvariant and skew-convex metric; it makes the semigroup into a Hausdorff topological inverse semigroup if the norm is cyclically permutable. Conversely, we show that every Clifford monoid equipped with a right-subinvariant and skew-convex metric admits a norm for which the metric topology and the norm topology coincide. We characterize convergence of nets and show that Cauchy completeness implies conditional monotone completeness with respect to the natural partial order of the inverse semigroup.

math.GR↗

arXiv2022

A memo on bornologies and size functions

We recall the notion of abstract bornology, and connect it with topological spaces and size functions. As a generalization of measures of non-compactness, we show how every size function can be mapped to a maxitive measure.

math.GM↗

arXiv2022

Galois connections between closure spaces

Galois connections were introduced by Ore and have proved useful in a wide variety of mathematical areas. While Galois connections play on the ground of posets (or more generally of quasiordered sets or qosets), we extend this notion to that of closure spaces.

math.GM↗

arXiv2022

Transporting continuity properties from a poset to its subposets

We identify two key conditions that a subset $A$ of a poset $P$ may satisfy to guarantee the transfer of continuity properties from $P$ to $A$. We then highlight practical cases where these key conditions are fulfilled. Along the way we are led to consider subsets of a given poset $P$ whose way-below relation is the restriction of the way-below relation of $P$, which we call way-below preserving subposets. As an application, we show that every conditionally complete poset with the interpolation property contains a largest continuous way-below preserving subposet. Most of our results are expressed in the general setting of Z theory, where Z is a subset system.

math.CO↗

arXiv2017

Enriched closure spaces as a novel framework for domain theory

We propose a generalization of continuous lattices and domains through the concept of enriched closure space, defined as a closure space equipped with a preclosure operator satisfying some compatibility conditions. In this framework we are able to define a notion of way-below relation; an appropriate definition of continuity then naturally follows. Characterizations of continuity of the enriched closure space and necessary and sufficient conditions for the interpolation property are proved. We also draw a link between continuity and the possibility for the subsets that are open with respect to the preclosure operator to form a topology.

cs.LO↗

arXiv2015

Representation of maxitive measures: an overview

Idempotent integration is an analogue of Lebesgue integration where $σ$-maxitive measures replace $σ$-additive measures. In addition to reviewing and unifying several Radon--Nikodym like theorems proven in the literature for the idempotent integral, we also prove new results of the same kind.

math.FA↗

arXiv2014

Convexities on ordered structures have their Krein--Milman theorem

We show analogues of the classical Krein-Milman theorem for several ordered algebraic structures, especially in a semilattice (non-linear) framework. In that case, subsemilattices are seen as convex subsets, and for our proofs we use arguments from continuous lattice theory and abstract convexity theory.

math.FA↗

arXiv2013

A memo on chains and their topologies

We summarize some facts on chains (totally ordered sets), from an order-theoretic and from a topological point of view. We highlight the fact that many classical theorems that are true for partially ordered sets under some completeness assumption remain true for chains without any kind of completeness.

math.GN↗

arXiv2013

Domain theory and mirror properties in inverse semigroups

Inverse semigroups are a class of semigroups whose structure induces a compatible partial order. This partial order is examined so as to establish mirror properties between an inverse semigroup and the semilattice of its idempotent elements, such as continuity in the sense of domain theory.

math.RA↗

arXiv2013

How regular can maxitive measures be?

We examine domain-valued maxitive measures defined on the Borel subsets of a topological space. Several characterizations of regularity of maxitive measures are proved, depending on the structure of the topological space. Since every regular maxitive measure is completely maxitive, this yields sufficient conditions for the existence of a cardinal density. We also show that every outer-continuous maxitive measure can be decomposed as the supremum of a regular maxitive measure and a maxitive measure that vanishes on compact subsets under appropriate conditions.

math.GN↗

arXiv2013

Two-valued sigma-maxitive measures and Mesiar's hypothesis

We reformulate Mesiar's hypothesis [Possibility measures, integration and fuzzy possibility measures, Fuzzy Sets and Systems 92 (1997) 191-196], which as such was shown to be untrue by Murofushi [Two-valued possibility measures induced by $σ$-finite $σ$-additive measures, Fuzzy Sets and Systems 126 (2002) 265-268]. We prove that a two-valued $σ$-maxitive measure can be induced by a $σ$-additive measure under the additional condition that it is $σ$-principal.

math.FA↗

arXiv2013

Pruning a poset with veins

We recall some abstract connectivity concepts, and apply them to special chains in partially ordered sets, called veins, that are defined as order-convex chains that are contained in every maximal chain they meet. Veins enable us to define a new partial order on the same underlying set, called the pruning order. The associated pruned poset is simpler than the initial poset, but irreducible, coirreducible, and doubly-irreducible elements are preserved by the operation of pruning.

cs.DM↗

arXiv2013

Pseudo-multiplications and their properties

We examine some properties of pseudo-multiplications, which are a special kind of associative binary relations defined on $\bar{\mathbb{R}}_+ \times \bar{\mathbb{R}}_+$.

math.RA↗

arXiv2013

What is the role of continuity in continuous linear forms representation?

The recent extensions of domain theory have proved particularly efficient to study lattice-valued maxitive measures, when the target lattice is continuous. Maxitive measures are defined analogously to classical measures with the supremum operation in place of the addition. Building further on the links between domain theory and idempotent analysis highlighted by Lawson (2004), we investigate the concept of domain-valued linear forms on an idempotent (semi)module. In addition to proving representation theorems for continuous linear forms, we address two applications: the idempotent Radon--Nikodym theorem and the idempotent Riesz representation theorem. To unify similar results from different mathematical areas, our analysis is carried out in the general Z framework of domain theory.

math.GN↗

arXiv2013

The idempotent Radon--Nikodym theorem has a converse statement

Idempotent integration is an analogue of the Lebesgue integration where $σ$-additive measures are replaced by $σ$-maxitive measures. It has proved useful in many areas of mathematics such as fuzzy set theory, optimization, idempotent analysis, large deviation theory, or extreme value theory. Existence of Radon--Nikodym derivatives, which turns out to be crucial in all of these applications, was proved by Sugeno and Murofushi. Here we show a converse statement to this idempotent version of the Radon--Nikodym theorem, i.e. we characterize the $σ$-maxitive measures that have the Radon--Nikodym property.

math.FA↗

arXiv2011

A class of compact subsets for non-sober topological spaces

We define a class of subsets of a topological space that coincides with the class of compact saturated subsets when the space is sober, and with enough good properties when the space is not sober. This class is introduced especially in view of applications to capacity theory.

math.GN↗