Laura Cossu<\/strong> Outline<\/strong><\/p>\n Many 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><\/p>\n 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><\/p>\n Prerequisites:<\/span> Standard knowledge of (basic) algebraic structures and (binary) relations.<\/span><\/p>\n Duration of the course:<\/span> 10 hours.<\/span><\/p>\n Schedule<\/strong><\/p>\n – November 28, 16:00-18:00 (Aula B) Exam<\/strong><\/span><\/p>\n Written. Writing of a short essay.<\/span><\/p>\n References<\/span><\/strong><\/p>\n 1. L. Cossu and S. Tringali,<\/span> Abstract Factorization Theorems with Applications to Idempotent Factor<\/span>izations<\/span><\/strong>, e-print (Aug. 2021),<\/span> arXiv:2108.12379<\/span>.<\/span> Factorizations in monoids and rings Laura Cossu University of Graz Outline Many problems in algebra involve the decomposition of certain elements of a ring (or more generally of a monoid) into a product of certain other […]<\/p>\n","protected":false},"author":3006,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[2],"tags":[],"class_list":["post-1111","post","type-post","status-publish","format-standard","hentry","category-news"],"_links":{"self":[{"href":"https:\/\/dottorati.unica.it\/matematicaeinformatica\/wp-json\/wp\/v2\/posts\/1111","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/dottorati.unica.it\/matematicaeinformatica\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/dottorati.unica.it\/matematicaeinformatica\/wp-json\/wp\/v2\/types\/post"}],"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=1111"}],"version-history":[{"count":5,"href":"https:\/\/dottorati.unica.it\/matematicaeinformatica\/wp-json\/wp\/v2\/posts\/1111\/revisions"}],"predecessor-version":[{"id":1265,"href":"https:\/\/dottorati.unica.it\/matematicaeinformatica\/wp-json\/wp\/v2\/posts\/1111\/revisions\/1265"}],"wp:attachment":[{"href":"https:\/\/dottorati.unica.it\/matematicaeinformatica\/wp-json\/wp\/v2\/media?parent=1111"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/dottorati.unica.it\/matematicaeinformatica\/wp-json\/wp\/v2\/categories?post=1111"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/dottorati.unica.it\/matematicaeinformatica\/wp-json\/wp\/v2\/tags?post=1111"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}
\nUniversity of Graz<\/p>\n
\n– November 29, 16:00-18:00 (Aula A)
\n– November 30, 16:00-18:00 (Aula A)
\n– December 1, 16:00-18:00 (Aula A)
\n– December 2, 16:00-18:00 (Aula A)<\/p>\n
2. L. Cossu and S. Tringali,<\/span> Factorization under Local Finiteness Conditions<\/span><\/strong>, e-print (Aug. 2022), <\/span>arXiv:2208.05869<\/span>.<\/span>
3. A. Geroldinger and F. Halter-Koch,<\/span> Non-Unique Factorizations<\/strong>. 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><\/strong>, J. Algebra 602 (2022), 352\u2013<\/span>380.<\/span><\/p>\n","protected":false},"excerpt":{"rendered":"