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The Mathematics of Ivo Rosenberg (Dedicated to the Memory of Professor Ivo Rosenberg) Miguel Couceiro, Lucien Haddad, Maurice Pouzet

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The Mathematics of Ivo Rosenberg (Dedicated to the Memory of Professor Ivo Rosenberg) Miguel Couceiro, Lucien Haddad, Maurice Pouzet The mathematics of Ivo Rosenberg (Dedicated to the memory of Professor Ivo Rosenberg) Miguel Couceiro, Lucien Haddad, Maurice Pouzet To cite this version: Miguel Couceiro, Lucien Haddad, Maurice Pouzet. The mathematics of Ivo Rosenberg (Dedicated to the memory of Professor Ivo Rosenberg). ISMVL2019 - IEEE 49th International Symposium on Multiple-Valued Logic, May 2019, Fredericton, NB, Canada. pp.43-48. hal-02190088 HAL Id: hal-02190088 https://hal.inria.fr/hal-02190088 Submitted on 21 Jul 2019 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. The mathematics of Ivo Rosenberg Dedicated to the memory of Professor Ivo Rosenberg. Miguel Couceiro∗, Lucien Haddady, Maurice Pouzetz ∗ Université de Lorraine, CNRS, Inria, LORIA, F-54000 Nancy, France Email: miguel.couceiro@{loria,Inria}.fr y Department of Mathematics & Computer Science, Royal Military College of Canada, Kingston, Ontario, Canada Email: [email protected] z ICJ, Univ. Claude-Bernard Lyon 1, Villeurbanne, France and Dept. of Mathematics & Statistics, University of Calgary, Alberta, Canada Email: [email protected] Abstract—We survey some of the most well known results technic) in 2006 and the second from the Universität Rostock of Professor Ivo G. Rosenberg as well as present some new (Rostock University) in 2013. ones related to the study of maximal partial clones and their Ivo Rosenberg organized several scientific meetings in Mon- intersections. tréal and elsewhere, notably many sessions of the well known After a long struggle with Parkinson’s disease, yet hard “Séminaire de Mathématiques Supérieures de l’Université working and keeping a very kind spirit, Professor Ivo G. de Montréal" like, for example, the “24e session: Algèbre Rosenberg died on January 18, 2018. He was born in Brno universelle et relations" (Montréal, July 23–August 10, 1984), (Moravia, Czech Republic), on December 13, 1934. One and the “30e session “Algèbres et ordres" (Montréal, July 29– year before obtaining the doctoral degree, he published his August 9, 1991) as well as the “premier colloque montréalais Completeness Criterion [49] in 1965. In July 1966, Professor sur la combinatoire et l’informatique" (Montréal, April 27– Rosenberg received his doctoral degree on his very same May 2, 1987). wedding day. Together with his wife Vlasta (known as Loty to Together with Maurice Pouzet, he organized the conference all family friends), he moved to Saskatoon, Canada, in Septem- “Aspects of Discrete Mathematics and Computer Science" ber 1968. After the birth of their two children, Madeleine (Lyon, June 24–27, 1987) within the first “Entretiens du Centre (known as Misha) and Marc, the family moved to Montréal in Jacques-Cartier". The second “Entretiens", called “Discrete September 1971. Ivo held the position of researcher with the Mathematics and Computer Science" (Montréal, October 12– rank of Full Professor from 1971 to 1983 at the “Centre de 14, 1988), took place in Montréal the year after. It was recherches mathématiques de l’Université de Montréal", and organized by Ivo Rosenberg together with A. Achache, M. later he was appointed as a full Professor at the “Département Pouzet (Lyon) and G. Hahn (Montréal). Still within his col- de mathématiques et de statistique" from 1983 to August 2011. laboration with French colleagues, Ivo organized a French- Ivo retired in August 2011. Quebec project “Ordered sets and their applications" from Professor Ivo G. Rosenberg was an eminent scholar, brilliant 1984 to 1986. mathematician, and one of the leading experts in universal On the other hand, Ivo Rosenberg organized several col- algebra and discrete mathematics. His huge impact and con- loquia such as (1) Algebraic, Extremal and Metric Combina- tributions to these areas of mathematics are very hard to assess torics, (1986) with M. Deza and P. Frankl (Proceedings pub- due to the great extension and ramifications of his works. lished in London Mathematical Society Lecture Notes Series, According to [38], Ivo Rosenberg wrote more than 200 papers 131), (2) Algebras and Orders (1991) with Gert Sabidussi and several books, and the number of his co-authors exceeds (Proceedings published in Nato Advanced Study Institute 55, among which are the three authors of this paper. Series, Math. and Phys. Sci., 389), and (3) Structural Theory Professor Rosenberg contributed to more than 50 conference of Automata, Semigroups and Universal Algebra, (2003), with papers in the various International Symposia on Multiple- V. B. Kudryavtsev (Proceedings published in Nato Science Valued Logic. Some of his contributions to these symposia Series, Math. Phys. and Chem. 207). are listed and discussed in [38]. He was extremely generous Several of his students became prominent mathematicians towards his colleagues and provided great hospitality both and computer scientists, to mention a few: Anne Fearnley, personally and through his position at the University of Jens-Uwe Grabowski, Lucien Haddad, Simone Hazan, Sebas- Montréal. He wrote numerous research reports and doctoral tian Kerkhoff, Gérard Kientega, Benoît Larose, Maxime Lau- thesis reports. zon, Florence Magnifo, Qinghe Sun, Bogdan Szczepara and For his outstanding research and contributions to mathemat- Calvin Wuntcha. The main goal of this survey is not to give ics, Professor Rosenberg received two honorary doctorates: an exhaustive account of Professor Rosenberg’s contributions, the first from the Technische Universität Wien (Vienna Poly- but rather to give a very brief survey of some of his most influential works in mathematics. For further information see, where a[i] (a 2 f0; 1; : : : ; hm − 1g) denotes the i-th digit e.g., [24], [28], [38]. in the h-adic expansion I. CLONES a = a[m−1] · hm−1 + a[m−2] · hm−2 + ··· + a[1] · h + a[0]; A. Completeness in Multiple-Valued Logic 6)a homomorphic inverse image of an h-ary relation ρ0 on Professor Ivo G. Rosenberg has been extremely active in re- k0, if there is a surjective mapping q : k ! k0 such that search and became one of the world’s leading mathematicians 0 in universal algebra and discrete mathematics. He became (a1; : : : ; ah) 2 ρ () (q(a1); : : : ; q(ah)) 2 ρ known in the 70’s for his general completeness criterion in multiple-valued logic. It was first presented in 1965 [49] and for all a1; : : : ; ah 2 k, the full proof was published in [50]. Rosenberg’s completeness 7) h-universal, if ρ is a homomorphic inverse image of an criterion is one of the most fundamental results in multiple h-ary elementary relation. valued logic and universal algebra, and its impact is impossible Furthermore, we denote by to estimate given the ever increasing number of research • Ck the set of all central relations on k; papers that followed it. h • C the set of all h-ary central relations on k; Let k := f0; 1; : : : ; k−1g, Op(k) be the set of all operations k • Uk the set of all non-trivial equivalence relations on k; on k and let Πk be the set of all projections on k. For X ⊆ • P the set of all fixed-point-free permutations on k Op(k), we denote by Clone(X) the set of all operations on k;p consisting of cycles of the same prime length p ; k that can be realized by composing functions from X [ Πk. 0 0 • S := fs j s 2 P g, where s := f(x; s(x)) j x 2 A set X ⊆ Op(k) is a clone if Clone(X) = X and X is said k;p k;p kg is the graph of s; to be complete if Clone(X) = Op(k). Thus X is complete • S := SfS j p is a prime divisor of kg; if and only if any function f : kn ! k can be realized by k k;p • M the set of all order relations on k with a least and a composing functions from X [ Π . For a relation ρ on k, let k k greatest element; Pol (ρ) be the set of all functions f 2 Op(k) preserving ρ. ? • M the set of all lattice orders on k; As a closure system, the set of all clones constitutes a lattice k • L the set of all prime affine relations on k; under C ^ C0 = C \ C0 and C _ C0 = Clone(C [ C0), and k • Bk the set of all h-universal relations, 3 ≤ h ≤ k − 1. where Op(k) and Πk are its greatest and least elements. It is also well known that this lattice is atomic and co-atomic, and Endowed with this terminology and notation, we can now its atoms and co-atoms are referred to as minimal and maximal state Rosenberg’s celebrated classification of maximal clones clones, respectively. on k. For the purpose of presenting Rosenberg’s classification of Theorem 1. ([49], [50]) Let k ≥ 2. Every proper clone on all maximal clones over k, we recall some families of relations k is contained in a maximal one. Moreover a clone M is a on k. For 1 ≤ h ≤ k, let maximal clone over k if and only if M = Pol ρ for some h relation ρ 2 Ck [Mk [Sk [Uk [Lk [Bk. ιk := f(a1; : : : ; ah) 2 k j ai = aj for some 1 ≤ i < j ≤ hg: Since every clone C 6= Op(k) is contained in a maximal Let ρ be an h-ary relation on k and denote by Sh the set of clone on k, we have the following completness criterion.
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