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1. A weaker notion of the finite factorization property

   Jiang, Henry; Kanungo, Shihan; Kim, Harry (April 2024, Communications of the Korean Mathematical Society)

   An (additive) commutative monoid is called atomic if every given non-invertible element can be written as a sum of atoms (i.e., irreducible elements), in which case, such a sum is called a factorization of the given element. The number of atoms (counting repetitions) in the corresponding sum is called the length of the factorization. Following Geroldinger and Zhong, we say that an atomic monoid M is a length-finite factorization monoid if each b ∈ M has only finitely many factorizations of any prescribed length. An additive submonoid of ℝ≥0 is called a positive monoid. Factorizations in positive monoids have been actively studied in recent years. The main purpose of this paper is to give a better understanding of the non-unique factorization phenomenon in positive monoids through the lens of the length-finite factorization property. To do so, we identify a large class of positive monoids which satisfy the length-finite factorization property. Then we compare the length-finite factorization property to the bounded and the finite factorization properties, which are two properties that have been systematically investigated for more than thirty years. 

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2. Consistency of Relations over Monoids

   https://doi.org/10.1145/3651608  

   Atserias, Albert; Kolaitis, Phokion G (May 2024, Proceedings of the ACM on Management of Data)

   The interplay between local consistency and global consistency has been the object of study in several different areas, including probability theory, relational databases, and quantum information. For relational databases, Beeri, Fagin, Maier, and Yannakakis showed that a database schema is acyclic if and only if it has the local-to-global consistency property for relations, which means that every collection of pairwise consistent relations over the schema is globally consistent. More recently, the same result has been shown under bag semantics. In this paper, we carry out a systematic study of local vs. global consistency for relations over positive commutative monoids, which is a common generalization of ordinary relations and bags. Let K be an arbitrary positive commutative monoid. We begin by showing that acyclicity of the schema is a necessary condition for the local-to-global consistency property for K-relations to hold. Unlike the case of ordinary relations and bags, however, we show that acyclicity is not always sufficient. After this, we characterize the positive commutative monoids for which acyclicity is both necessary and sufficient for the local-to-global consistency property to hold; this characterization involves a combinatorial property of monoids, which we call the transportation property. We then identify several different classes of monoids that possess the transportation property. As our final contribution, we introduce a modified notion of local consistency of K-relations, which we call pairwise consistency up to the free cover. We prove that, for all positive commutative monoids K, even those without the transportation property, acyclicity is both necessary and sufficient for every family of K-relations that is pairwise consistent up to the free cover to be globally consistent. 

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3. Algebraic Systems Motivated by DNA Origami

   Garrett, J.; Jonoska, N.; Saito, M (January 2019, Algebraic Informatics)

   Ćirić, M.; Droste, M.; Pin, JÉ. (Ed.)

   We initiate an algebraic approach to study DNA origami structures. We identify two types of basic building blocks and describe a DNA origami structure by their composition. These building blocks are taken as generators of a monoid, called the origami monoid, and motivated by the well studied Temperley-Lieb algebras, we identify a set of relations that characterize the origami monoid. We present several observations about Green’s relations for the origami monoid and study the relations to a direct product of Jones monoids, which is a morphic image of an origami monoid. 

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4. Products and powers of principal symmetric ideals

   https://doi.org/10.1142/S0219498825503207  

   Dannetun, Eric; Fang, Bruce; Formenti, Riccardo; Gao, Bo Y; Geraci, Juliann; Kogel, Ross; Li, Yuelin; Mandal, Shreya; Rupasinghe, Vinuge; Seceleanu, Alexandra; et al (July 2024, Journal of Algebra and Its Applications)

   Principal symmetric ideals were recently introduced by Harada et al. in [The minimal free resolution of a general principal symmetric ideal, preprint (2023), arXiv:2308.03141], where their homological properties are elucidated. They are ideals generated by the orbit of a single polynomial under permutations of variables in a polynomial ring. In this paper, we determine when a product of two principal symmetric ideals is principal symmetric and when the powers of a principal symmetric ideal are again principal symmetric ideals. We characterize the ideals that have the latter property as being generated by polynomials invariant up to a scalar multiple under permutation of variables. Recognizing principal symmetric ideals is an open question for the purpose of which we produce certain obstructions. We also demonstrate that the Hilbert functions of symmetric monomial ideals are not all given by symmetric monomial ideals, in contrast to the non-symmetric case. 

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5. On the fixed part of pluricanonical systems for surfaces

   https://doi.org/10.1002/mana.202200088  

   Liu, Jihao; Xie, Lingyao (February 2023, Mathematische Nachrichten)

   Abstract We show that defines a birational map and has no fixed part for some bounded positive integermfor any ‐lc surfaceXsuch that is big and nef. For every positive integer , we construct a sequence of projective surfaces , such that is ample, for everyi, , and for any positive integerm, there existsisuch that has nonzero fixed part. These results answer the surface case of a question of Xu. 

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