{"id":1065,"date":"2022-02-25T19:30:03","date_gmt":"2022-02-25T18:30:03","guid":{"rendered":"https:\/\/dottorati.unica.it\/matematicaeinformatica\/?page_id=1065"},"modified":"2023-12-12T13:00:49","modified_gmt":"2023-12-12T12:00:49","slug":"courses-2022","status":"publish","type":"page","link":"https:\/\/dottorati.unica.it\/matematicaeinformatica\/courses\/courses-2022\/","title":{"rendered":"Courses 2022"},"content":{"rendered":"

Dal 28 novembre 2022 al 02 dicembre 2022 la Dott.ssa Laura Cossu, University of Graz, terra’ un corso di dottorato su: Factorizations in monoids and rings<\/span><\/p>\n


\nMany problems in algebra involve the decomposition of certain elements of a ring (or more generally<\/span>
of a monoid) into a product of certain other elements (hereinafter generically referred to as building<\/span>
blocks) that are in some sense minimal. The classical theory of factorization investigates factorizations<\/span>
in which the building blocks are atoms, i.e., non-unit elements of a monoid that are not products of<\/span>
two non-units. For example, it is well known that every non-zero non-unit of a Dedekind domain (more<\/span>
generally, of a Noetherian domain) can be written as a finite product of atoms and that in general<\/span>
such decompositions are not unique. On the other hand, examples of factorizations that lie beyond the<\/span>
scope of the classical theory include additive decompositions into multiplicative units in rings; cyclic<\/span>
decompositions of permutations in the symmetric group of degree n; idempotent factorizations of the<\/span>
\u201csingular elements\u201d of a monoid; and so on.<\/span>
After an introductory overview of the history and main results in the classical framework, in this<\/span>
course, we combine the language of monoids and preorders (partial orders that are not necessarily an-<\/span>
tisymmetric) to make first steps towards the construction of a \u201cunified theory of factorization\u201d.<\/span> In<\/span>
particular, we prove an abstract existence theorem that recover, among others, a classical theorem of<\/span>
Cohn on atomic factorizations in cancellative monoids, a classical result by Anderson and Valdes-Leon<\/span>
on \u201cirreducible factorizations\u201d in commutative monoids, but also a theorem by Erdos on idempotent fac-<\/span>
torizations of square singular matrices over fields. We also introduce a notion of \u201cminimal factorization\u201d,<\/span>
suitable for the new abstract setting, and we qualify and quantify the related non-uniqueness properties<\/span>
in specific cases but also in some generality. Examples will help to motivate and illustrate the theory.<\/span>
Prerequisites:<\/span> Standard knowledge of (basic) algebraic structures and (binary) relations.<\/span>
Duration of the course:<\/span> 10 hours.<\/span>
Exam:<\/span> Written. Writing of a short essay.<\/span>
References<\/span>
[1] L. Cossu and S. Tringali,<\/span> Abstract Factorization Theorems with Applications to Idempotent Factor-<\/span>
izations<\/span>, e-print (Aug. 2021),<\/span> arXiv:2108.12379<\/span>.<\/span>
[2] L. Cossu and S. Tringali,<\/span> Factorization under Local Finiteness Conditions<\/span>, e-print (Aug. 2022),<\/span>
arXiv:2208.05869<\/span>.<\/span>
[3] A. Geroldinger and F. Halter-Koch,<\/span> Non-Unique Factorizations. Algebraic, Combinatorial and An-<\/span>
alytic Theory<\/span>, Pure Appl. Math.<\/span> 278<\/span>, Chapman & Hall\/CRC, Boca Raton (FL), 2006.<\/span>
[4] S. Tringali,<\/span> An Abstract Factorization Theorem and Some Applications<\/span>, J. Algebra 602 (2022), 352\u2013<\/span>
380.<\/span>
1<\/span><\/p>\n

Il calendario delle lezioni \u00e8 il seguente: Luned\u00ec 28 novembre dalle 16:00 alle 18:00 Aula B Marted\u00ec 29 novembre dalle 16:00 alle 18:00 Aula A Mercoled\u00ec 30 novembre dalle 16:00 alle 18:00 Aula A Gioved\u00ec 1 dicembre dalle 16:00 alle 18:00 Aula A Venerd\u00ec 2 dicembre dalle 16:00 alle 18:00 Aula A<\/p>\n

 <\/p>\n

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Dal 20 al 30 giugno i Visiting Professors M. Lucia e P. Sicbaldi terranno\u00a0un\u00a0corso intitolato\u00a0“Geometric\u00a0Analysis”\u00a0per\u00a0un\u00a0totale\u00a0di 20 ore di lezione (dieci ciascuno). Le informazioni si trovano al link: https:\/\/www.unica.it\/unica\/page\/it\/antonio_greco_avs_geometric_analysis<\/a><\/p>\n

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 <\/p>\n","protected":false},"excerpt":{"rendered":"

Dal 28 novembre 2022 al 02 dicembre 2022 la Dott.ssa Laura Cossu, University of Graz, terra’ un corso di dottorato su: Factorizations in monoids and rings Many problems in algebra involve the decomposition of certain elements of a ring (or more generallyof a monoid) into a product of certain other elements (hereinafter generically referred to as buildingblocks) that are in some sense minimal. The classical theory of factorization investigates factorizationsin which the building blocks are atoms, i.e., non-unit elements of a monoid that are not products oftwo non-units. For example, it is well known that every non-zero non-unit of a […]<\/a><\/p>\n","protected":false},"author":3006,"featured_media":0,"parent":161,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"footnotes":""},"class_list":["post-1065","page","type-page","status-publish","hentry","post-seq-1","post-parity-odd","meta-position-corners","fix"],"_links":{"self":[{"href":"https:\/\/dottorati.unica.it\/matematicaeinformatica\/wp-json\/wp\/v2\/pages\/1065","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/dottorati.unica.it\/matematicaeinformatica\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/dottorati.unica.it\/matematicaeinformatica\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/dottorati.unica.it\/matematicaeinformatica\/wp-json\/wp\/v2\/users\/3006"}],"replies":[{"embeddable":true,"href":"https:\/\/dottorati.unica.it\/matematicaeinformatica\/wp-json\/wp\/v2\/comments?post=1065"}],"version-history":[{"count":13,"href":"https:\/\/dottorati.unica.it\/matematicaeinformatica\/wp-json\/wp\/v2\/pages\/1065\/revisions"}],"predecessor-version":[{"id":1273,"href":"https:\/\/dottorati.unica.it\/matematicaeinformatica\/wp-json\/wp\/v2\/pages\/1065\/revisions\/1273"}],"up":[{"embeddable":true,"href":"https:\/\/dottorati.unica.it\/matematicaeinformatica\/wp-json\/wp\/v2\/pages\/161"}],"wp:attachment":[{"href":"https:\/\/dottorati.unica.it\/matematicaeinformatica\/wp-json\/wp\/v2\/media?parent=1065"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}