Factorizations in upper triangular matrices over information semialgebras

[摘要] An integral domain (or a commutative cancellative monoid) is atomic if every nonzero nonunit element factors into atoms, and it satisfies the ACCP if every ascending chain of principal ideals eventually stabilizes. The interplay between these two properties has been investigated since the 1970s. An atomic domain (or monoid) satisfies the finite factorization property (FFP) if every element has only finitely many factorizations, and it satisfies the bounded factorization property (BFP) if for each element there is a bound for the number of atoms in each of its factorizations. These two properties have been systematically studied since they were introduced by Anderson, Anderson, and Zafrullah in 1990. Noetherian domains satisfy the BFP, while Dedekind domains satisfy the FFP. It is well known that, for commutative cancellative monoids (and, in particular, for multiplicative monoids of integral domains), FFP double right arrow BFP double right arrow ACCP double right arrow atomic. For n >= 2, we show that each of these four properties transfers back and forth between an information semialgebra S (certain commutative cancellative semiring) and the multiplicative monoid T-n(S)(center dot) consisting of n x n upper triangular matrices over S. We also show that a similar transfer behavior takes place if one replaces T-n(S)(center dot) by its submonoid U-n(S) consisting of upper triangular matrices with units along their main diagonals. As a consequence, we find that the atomic chain FFP double right arrow BFP double right arrow ACCP double right arrow atomic also holds for the two classes comprising the noncommutative monoids T-n(S)(center dot) and U-n(S). Finally, we construct various rational information semialgebras to verify that, in general, none of the established implications is reversible. (C) 2020 Elsevier Inc. All rights reserved.

[发布日期] 2020-11-15

[关键词] Puiseux semirings;Information semialgebras;Upper triangular matrices;Factorizations;Atomicity;ACCP;BFM;FFM;HFM