Il calendario delle lezioni è il seguente: Lunedì 28 novembre dalle 16:00 alle 18:00 Aula B Martedì 29 novembre dalle 16:00 alle 18:00 Aula A Mercoledì 30 novembre dalle 16:00 alle 18:00 Aula A Giovedì 1 dicembre dalle 16:00 alle 18:00 Aula A Venerdì 2 dicembre dalle 16:00 alle 18:00 Aula A

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 elements (hereinafter generically referred to as building

blocks) that are in some sense minimal. The classical theory of factorization investigates factorizations

in which the building blocks are atoms, i.e., non-unit elements of a monoid that are not products of

two non-units. For example, it is well known that every non-zero non-unit of a Dedekind domain (more

generally, of a Noetherian domain) can be written as a finite product of atoms and that in general

such decompositions are not unique. On the other hand, examples of factorizations that lie beyond the

scope of the classical theory include additive decompositions into multiplicative units in rings; cyclic

decompositions of permutations in the symmetric group of degree n; idempotent factorizations of the

“singular elements” of a monoid; and so on.

After an introductory overview of the history and main results in the classical framework, in this

course, we combine the language of monoids and preorders (partial orders that are not necessarily an-

tisymmetric) to make first steps towards the construction of a “unified theory of factorization”. In

particular, we prove an abstract existence theorem that recover, among others, a classical theorem of

Cohn on atomic factorizations in cancellative monoids, a classical result by Anderson and Valdes-Leon

on “irreducible factorizations” in commutative monoids, but also a theorem by Erdos on idempotent fac-

torizations of square singular matrices over fields. We also introduce a notion of “minimal factorization”,

suitable for the new abstract setting, and we qualify and quantify the related non-uniqueness properties

in specific cases but also in some generality. Examples will help to motivate and illustrate the theory.

Prerequisites: Standard knowledge of (basic) algebraic structures and (binary) relations.

Duration of the course: 10 hours.

Exam: Written. Writing of a short essay.

References

[1] L. Cossu and S. Tringali, Abstract Factorization Theorems with Applications to Idempotent Factor-

izations, e-print (Aug. 2021), arXiv:2108.12379.

[2] L. Cossu and S. Tringali, Factorization under Local Finiteness Conditions, e-print (Aug. 2022),

arXiv:2208.05869.

[3] A. Geroldinger and F. Halter-Koch, Non-Unique Factorizations. Algebraic, Combinatorial and An-

alytic Theory, Pure Appl. Math. 278, Chapman & Hall/CRC, Boca Raton (FL), 2006.

[4] S. Tringali, An Abstract Factorization Theorem and Some Applications, J. Algebra 602 (2022), 352–

380.