Factorizations in upper triangular matrices over information semialgebras

Abstract

An integral domain (or a commutative cancellative monoid) is atomic if every nonzero nonunit element is the product of irreducibles, 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 common bound for the number of atoms in each of its factorizations. These two properties have been systematically studied since being 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 (in particular, integral domains) FFP $\Rightarrow$ BFP $\Rightarrow$ ACCP $\Rightarrow$ atomic. For $n \ge 2$, we show that each of these four properties transfers back and forth between an information semialgebras $S$ (i.e., a commutative cancellative semiring) and their multiplicative monoids $T_n(S)^\bullet$ of $n \times n$ upper triangular matrices over~$S$. We also show that a similar transfer behavior takes place if one replaces $T_n(S)^\bullet$ by the submonoid $U_n(S)$ consisting of unit triangular matrices. As a consequence, we find that the chain FFP $\Rightarrow$ BFP $\Rightarrow$ ACCP $\Rightarrow$ atomic also holds for the classes comprising the noncommutative monoids $T_n(S)^\bullet$ and $U_n(S)$. Finally, we construct various rational information semialgebras to verify that, in general, none of the established implications is reversible.