# Xavier Mary

MODAL'X, UMR CNRS 9023, UPL, Université Paris Nanterre

Bat. Allais, Bureau E10
200 avenue de la République,
92000 Nanterre, France

E-mail: xavier.mary at parisnanterre.fr

## Research Interests

• Generalized Inverses in semigroups and rings

• Semigroups - structure theory, Green's relations, S-acts

• Rings - Chains of idempotents, perspective elements and rings, special clean elements, general (non-unital) rings

• Hilbert subspaces and Reproducing Kernel Hilbert Spaces, Subdualities, Krein Spaces

• Determinantal processes and sampling

## Workshop Generalized Inverses, Semigroups and Rings" May 30 - June 2, 2022

The Workshop will take place at Paris Nanterre University, at the occasion of the Defense of my Professorial Thesis (June 1st 2022).
We will discuss recent progresses on the following subjects:
• Generalized inverses - Inverses along an element, (b,c)-inverses, group inverses, Drazin inverses, Cline's formula, Jacobson lemma, Reverse Order Laws
• Structure theory of semigroups - by means of Green's (extended) relations, generalized inverses, and chains of idempotents
• Structure of S-acts
• Elementwise approach to ring and module theory - Perspective and special clean elements, generalizations to non-unital rings
Participants:
• Victoria Gould (York University, Great Britain)
• André Leroy (Université d’Artois, France)
• Laszlo Marki (Renyi Institute, Hungaria)
• Xavier Mary (Unversité Paris Nanterre, France)
• Dijana Mosic (Nis University, Serbia)
• Pedro Patricio (Universidad do Minho, Braga, Portugal)

The Webpage of the Workshop is here.

## Jury

• Victoria Gould (York University, Great Britain) -- Thesis Reporter
• André Leroy (Université d’Artois, France) -- Thesis Reporter
• Christian Léonard (Université de Nanterre, France) -- Local Reffering Examiner
• Laszlo Marki (Renyi Institute, Hungaria) -- Examiner, President of the Jury
• Dijana Mosic (Nis University, Serbia) -- Thesis Reporter
• Pedro Patricio (Universidad do Minho, Braga, Portugal) -- Examiner

## Memoir reports

• Report of Dijana Mosic (Nis University, Serbia) on the generalized inverses part
• Report of Victoria Gould (York University, Great Britain) on the semigroup part
• Report of André Leroy (Université d’Artois, France) on the ring part
• Report of Christian Léonard (Université d’Artois, France)

## Preprints

• D. Khurana, X.Mary, and P.P. Nielsen. Idempotent chains and bounded generation of SL2. Submitted.
We investigate connections between association chains of idempotents and bounded generation of SL2 by elementary matrices. This enables us to construct many classes of matrix rings that have universal bounds on the length of association chains. In particular, many matrix rings have the property that every regular element is special clean. We also improve the usual criterion for checking perspectivity via association chains.
• X. Mary. Rings with transitive chaining of idempotents. Submitted to Proceedings of the NCRA, VII. (Preprint Version)
We prove that transitivity of strong or weak 2-chaining of idempotents is equivalent to centrality of idempotents modulo the Jacobson radical.
• X. Mary. Special clean elements, perspective elements and perspective rings. To appear in J. Algebra Appl. (Preprint Version)
Motivated by the idea of perspectivity of rings and modules, we introduce perspective elements of a ring. We notably prove that perspective elements form a proper subset of special clean elements, and that perspectivity of elements is a left-right symmetric property. An equational characterization of perspective elements is also given and examples are provided.
• X. Mary. n-chained semigroups and n/2-perspective modules and rings. To appear in Comm. Algebra. (Preprint Version)
We introduce and study a new class of modules and rings we call n/2-perspective, which can either be described in terms of perspective direct summands, associate idempotents, or generalized inverses. When n is small (n<3), we recover existing classes of modules and rings: (endo)abelian, strongly IC and perspective ones. And 3/2-perspective rings are characterized by all their regular elements being special clean. Standard constructions are also discussed and examples are provided.

## Publications

[28] X. Mary and P.P. Nielsen, IC rings and transitivity of perspectivity. J. Algebra Appl.,  2250151 (2021). /---/ Preprint Version

We construct an example of an IC ring where perspectivity is transitive, but not all isomorphic idempotents are perspective. We also develop new criteria for checking perspectivity of idempotents in rings.

[27] X. Mary, (b, c)-inverse, inverse along and element, and the Schützenberger category of a semigroup. Categories Gen. Algebraic Struct. with Appl., 15(1):255–272 (2021).

We prove that the (b, c)-inverse and the inverse along an element in a semigroup are actually genuine inverse when considered as morphisms in the Schützenberger category of a semigroup. Applications to the Reverse Order Law are given.

[26] X. Mary and P. Patricio, Characterizations of special clean elements and applications. Filomat, 34(14): 4847-4860 (2020)

We prove that special clean decompositions of a given element of a ring are in one-to-one correspondence with the set of solutions of a simple equation in a corner ring.

We characterize clean elements in unital and general rings by means of outer inverses. Some special cases, such as both clean and unit-regular elements, or strongly clean elements, are discussed. As an application, we also derive new characterizations of strongly regular elements.

[24] X. Mary and P. Patricio, Group-regular rings. Filomat, 33(11):3551-3560 (2019)

We propose different generalizations of unit-regularity of elements in general rings (non necessarily unital rings). We then study general rings for which all elements have these properties. We notably compare them with unit-regular ideals and general rings with stable range one.

[23] A. Guterman, X. Mary and P. Shteyner, On Hartwig-Nambooripad orders, Semigroup Forum, 98 (2019), no. 1, 64–74

We obtain several equivalent characterizations for the Hartwig–Nambooripad order based on outer inverses. We also introduce new partial orders on arbitrary semigroups extending these regular semigroup orders. On epigroups these new partial orders admit characterizations in terms of outer inverses.

In this article, recent results about point processes are used in sampling theory. Precisely, we define and study a new class of sampling designs: determinantal sampling designs. The law of such designs is known, and there exists a simple selection algorithm. We compute exactly the variance of linear estimators constructed upon these designs by using the first and second order inclusion probabilities. Moreover, we obtain asymptotic and finite sample theorems. We construct explicitly fixed size determinantal sampling designs with given first order inclusion probabilities. We also address the search of optimal determinantal sampling designs.

We describe a class of semigroup biacts that is analogous to the class of completely simple semigroups,and prove a structure theorem for those biacts that is analogous to the Rees--Sushkevitch Theorem.Precisely, we describe stable, J-simple biacts in terms of wreath products, translations of completely simple semigroups, biacts over endomorphism monoids of free G-acts,tensor products and matrix biacts. Applications to coproducts and left acts are given.

[20] A. Guterman, X. Mary and P. Shteyner, Partial orders based on inverses along an element, Journal of Mathematical Sciences 232 (2018), no. 6

The paper introduces and investigates partial orders that are finer than the minus partial order and are based on inverses along an element and other specific outer inverses. It turns out that in this way a number of classical partial orders can be equivalently defined.

We study extensions of Cline’s formula and Jacobson lemma for one-sided, commuting and bicommuting weak inverses, in semigroups and general rings. In particular, we provide various isomorphisms between the (one-sided, commuting and bicommuting) weak inverses of ab and those of ba.

[18] A. Chambaz, E. Joly and X. Mary, Targeted sampling using adaptative survey sampling, chapter 29 in Targeted Learning in Data Science, by S. Rose and M. J. van der Laan (Springer, 2018)

[17] R. Bardenet, F. Lavancier, X. Mary, and A. Vasseur, On a few statistical applications of determinantal point processes, ESAIM: Procs, 60 (2017) 180-202

We study semigroups that behave nicely with respect to a distinguished subset of idempotents E, both in terms of the extended Green’s relations KE and as unary semigroups. New structure theorems are given, notably in the case of central idempotents. Finally, the decomposition theorems are applied to the study of regular semigroups with particular generalized inverses.

[15] H. Zhu, J. Chen, P. Patricio and X. Mary, Centralizer’s applications to the inverse along an element, Appl. Math. Comp. 315 (2017) 27-33

In this paper, we firstly prove that the absorption law for one-sided inverses along an element holds, and derive the absorption law for the inverse along an element. We then obtain the absorption law for the inverse along different elements. Then, the reverse order law and the existence criterion for the inverse along an element are given by centralizers in a ring.

[14] X. Mary and P. Patricio, The group inverse of a product. Linear and Multilinear Algebra 64 (2016), no. 9, 1776-1784. /---/ Preprint Version

In this paper, we characterize the existence and give an expression of the group inverse of a product of two regular elements by means of a ring unit.

In this paper, we provide equivalent conditions for the two-sided reverse order law for the group inverse: (ab)# = b#a# and (ba)# = a#b#, in semigroups and rings. Moreover, we prove that under finiteness conditions, these conditions are also equivalent with the one-sided reverse order law (ab)# = b#a# .

Inverse semigroups and orthodox semigroups are either defined in terms of inverses, or in terms of the set of idempotents E(S). In this article, we study analogs of these semigroups defined in terms of inverses modulo Green’s relation H, or in terms of the set of completely regular elements H(S). Results are obtained both for the regular and the non-regular cases. We then study the interplays between these new classes of semigroups, as well as with various known classes notably of inverse, orthodox, E-solid and cryptic semigroups.

Using the recent notion of inverse along an element in a semigroup, and the natural partial order on idempotents, we study bicommuting generalized inverses and define a new inverse called natural inverse, that generalizes the Drazin inverse in a semigroup, but also the Koliha-Drazin inverse in a ring. In this setting we get a core decomposition similar to the nilpotent, Kato or Mbekhta decompositions. In Banach and Operator algebras, we show that the study of the spectrum is not sufficient, and use ideas from local spectral theory to study this new inverse.

[10] X. Mary and P. Patricio. Generalized inverses modulo H in semigroups and rings. Linear and Multilinear Algebra 61(8) (2013), 1130-1135. / Preprint Version

The definition of the inverse along an element was very recently introduced, and it contains known generalized inverses such as the group, Drazin and Moore–Penrose inverses. In this article, we first prove a simple existence criterion for this inverse in a semigroup, and then relate the existence of such an inverse in a ring to the ring units.

[9] X. Mary and P. Patricio. The inverse along a lower triangular matrix. Appl. Math. Comput 219 (3)(2012) 886-891. / Preprint Version

In this paper, we investigate the recently defined notion of inverse along an element in the context of matrices over a ring. Precisely, we study the inverse of a matrix along a lower triangular matrix, under some conditions.

(Reprint: Linear Algebra Appl. 438 (2013), no. 4, 1532–1540.)

We study generalized inverses on semigroups by means of Green’s relations. We first define the notion of inverse along an element and study its properties. Then we show that the classical generalized inverses (group inverse, Drazin inverse and Moore–Penrose inverse) belong to this class.

We discuss the notion of Moore-Penrose inverse in Kre?n spaces for both bounded and unbounded operators. Conditions for the existence of a Moore-Penrose inverse are given. We then investigate its relation with adjoint operators, and study the involutive Banach algebra . Finally applications to the Schur complement are given.

We prove that the converse of Theorem 9 in “On generalized inverses in C^*-algebras” by Harte and Mbekhta (Studia Math. 103 (1992)) is indeed true : a element $a$ of a unital ring is group invertible if and only if n $aA = a^2A$ and $Aa = Aa^2$.

[5] S. Canu, C. S. Ong et X. Mary, Splines with Non-Positive Kernels. In Proceedings of the 5th ISAAC Congress , World Scientific, (2006), 1-10.

Non parametric regression methods can be presented in two main clusters. The one of smoothing splines methods requiring positive kernels and the other one known as Nonparametric Kernel Regression allowing the use of non positive kernels such as the Epanechnikov kernel. We propose a generalization of the smoothing spline method to include kernels which are still symmetric but not positive semi definite (they are called indefinite). The general relationship between smoothing splines, Reproducing Kernel Hilbert Spaces (RKHS) and positive kernels no longer exists with indefinite kernels. Instead the splines are associated with functional spaces called Reproducing Kernel Krein Spaces (RKKS) endowed with an indefinite inner product and thus not directly associated with a norm. Smoothing splines in RKKS have many of the interesting properties of splines in RKHS, such as orthogonality, projection and representer theorem.  We show that smoothing splines can be defined in RKKS as the regularized solution of the interpolation problem. Since no norm is available in an RKKS, Tikhonov regularization cannot be defined. Instead, we propose the use of conjugate gradient type iterative methods, with early stopping as a regularization mechanism. Several iterative algorithms are collected which can be used to solve the optimization problems associated with learning in indefinite spaces. Some preliminary experiments with indefinite kernels for spline smoothing reveal the computational efficiency of this approach.

We present a new theory of dual systems of vector spaces that extends the existing notions of reproducing kernel Hilbert spaces and Hilbert subspaces. In this theory, kernels (understood as operators rather than kernel functions) need not be positive or self-adjoint. These dual systems, called subdualities, enjoy many properties similar to those of Hilbert subspaces and include the notions of Hilbert subspaces or Kreîn subspaces as particular cases. Some applications to Green operators or invariant subspaces are given.

[3] A. Rakotomamonjy, X. Mary et S. Canu, Non-parametric regression with wavelet kernels. Appl. Stoch. Models Bus. Ind. 21 (2005), no. 2, 153-163.

This paper introduces a method to construct a reproducing wavelet kernel Hilbert spaces for non-parametric regression estimation when the sampling points are not equally spaced. Another objective is to make high-dimensional wavelet estimation problems tractable. It then provides a theoretical foundation to build reproducing kernel from operators and a practical technique to obtain reproducing kernel Hilbert spaces spanned by a set of wavelets. A multiscale approximation technique that aims at taking advantage of the multiresolution structure of wavelets is also described. Examples on toy regression and a real-world problem illustrate the effectiveness of these wavelet kernels. Copyright © 2005 John Wiley & Sons, Ltd

[2] C. S. Ong, X. Mary, S. Canu et A. J. Smola, Learning with Non-Positive Kernels. In Proceedings of the 21st International Conference on Machine Learning, ACM Press, (2004), 639-646.

In this paper we show that many kernel methods can be adapted to deal with indefinite kernels, that is, kernels which are not positive semidefinite. They do not satisfy Mercer’s condition and they induce associated functional spaces called Reproducing Kernel Kre??n Spaces (RKKS), a generalization of Reproducing Kernel Hilbert Spaces (RKHS).

[1] X. Mary, D. De Brucq et S. Canu, Sous-dualités et noyaux (reproduisants) associés. C. R. Math. Acad. Sci. Paris - Série I : Mathématiques 336/11 (2003), 949-954.

A way to study non-Gaussian measures is to generalize Schwartz's theory of Hilbertian subspaces to the non-Hilbertian case. We initiate here a new theory based upon a given duality rather than an Hilbertian structure. Hence, we develop the concept of subduality of a locally convex vector space and of its associated kernel. We show in particular that to any subduality is associated a unique kernel whose image is dense in the subduality. The image of a subdality under a weakly continuous linear application is also given, which enables the definition of a vector space structure over the set of subdualities given a equivalence relation. We then exhibit a canonical representative. Finally, we study the particular case of subdualities of endowed with the product topology.