Factorizations in upper triangular matrices over information semialgebras

Abstract

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 ⇒ BFP ⇒ ACCP ⇒ atomic. For $n\ge 2$, 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 $T_n{(S)}^{•}$ consisting of $n\times n$ upper triangular matrices over S. We also show that a similar transfer behavior takes place if one replaces $T_n{(S)}^{•}$ 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 ⇒ BFP ⇒ ACCP ⇒ atomic also holds for the two classes comprising the noncommutative monoids $T_n{(S)}^{•}$ and $U_n(S)$. Finally, we construct various rational information semialgebras to verify that, in general, none of the established implications is reversible.

Introduction

A factorization of an element in a cancellative monoid is a representation of that element as a formal product of atoms (i.e., irreducible elements). When every nonunit element has such a representation, the monoid is called atomic and, additionally, if such a representation is unique, the monoid is called a unique factorization monoid (or a UFM). An atomic monoid is called a finite factorization monoid (or an FFM) if every nonunit element has only finitely many factorizations, and it is called a bounded factorization monoid (or a BFM) if for each nonunit element there is a bound for the number of atoms (counting repetitions) in each of its factorizations. In addition, an atomic monoid is called a half-factorial monoid (or an HFM) if any two factorizations of the same nonunit element involve the same number of atoms (counting repetitions). Atomic monoids, monoids satisfying the ACCP, and HFMs have been systematically studied since the 1970s, while FFMs and BFMs have been studied since they were introduced in 1990 [1] in the context of integral domains. For any commutative monoid, the implications in Diagram (1.1) are rather straightforward to verify. On the other hand, it was illustrated in [1] that none of such implications is reversible in the class of integral domains.

Although most of the work on factorization theory has taken place in the context of commutative rings and monoids, factorization aspects on commutative semirings have also been considered. For instance, it was proved in [9] that the multiplicative monoids of the semirings ${\mathbb{R}}_{\ge 0}[x]$ and ${\mathbb{N}}_0[\tau ]$ (where τ is a reasonable quadratic algebraic integer) have full infinite elasticity (the elasticity, as a factorization-theoretic invariant, was introduced in [29] and has been studied extensively since then). On the other hand, certain factorizations of polynomials of ${\mathbb{N}}_0[x]$ were considered in [7], where a connection to non-unique factorization theory was pointed out. A more systematic investigation of factorizations in the multiplicative monoid of the semiring ${\mathbb{N}}_0[x]$ was recently carried out in [8]. In addition, various factorization invariants of the additive monoid of the semiring $\{p(r):p\in {\mathbb{N}}_0[x]\}$ (where $r\in {\mathbb{Q}}_{>0}$) were studied in [10]. Finally, certain arithmetic properties of commutative semigroup semirings were considered in [27].

Factorization theory has been significantly less developed in noncommutative settings, with much of the early work focusing primarily on characterizing when a given monoid is a UFM (see, for instance, [14, Chapter 3] and [15]). However, in recent years there has been more consideration given to arithmetic and factorization aspects of noncommutative rings and monoids. In particular, many factorization tools from commutative monoids and domains have been used in and adapted to various noncommutative algebraic structures, including rings of upper triangular nonnegative matrices [12], multiplicative monoids of matrices over chain semirings [13], maximal orders in central simple algebras [28], noncommutative finite factorization domains [6], noncommutative Krull monoids [17], and small cancellative categories [4].

In many cases, factorization aspects of noncommutative algebraic structures are conveniently investigated through the lens of transfer homomorphisms to easier-to-understand commutative objects. For example, in [2] the noncommutative monoids $T_n(R)$ of $n\times n$ upper triangular matrices over a commutative ring R are studied using transfer homomorphisms to products of commutative cancellative monoids, and in [4] various arithmetic aspects of noncommutative cancellative monoids are studied using transfer homomorphisms to their reduced abelianizations. By contrast, it was proved in [3] that for a reduced information semialgebra S (i.e., a commutative semiring whose additive and multiplicative monoids are both cancellative and reduced), there are no such transfer homomorphisms from the monoid $T_n{(S)}^{•}$ of regular elements of $T_n(S)$. Using other approaches, however, it was shown in [3] that $T_n{(S)}^{•}$ is atomic, after which some factorization invariants were computed. Factorizations in $T_n{(S)}^{•}$, which are not only far from trivial but also interesting by themselves, can be conveniently investigated once we have a good understanding of the factorization structure of S, as the following examples indicate.

Example 1.1

• It was proved in [3] that $T_n{({\mathbb{N}}_0)}^{•}$ is a BFM for every $n\in \mathbb{N}$ (clearly, the additive monoid ${\mathbb{N}}_0$ is a BFM). Let S be the additive submonoid of ${\mathbb{Q}}_{\ge 0}$ generated by the set $A=\{{(2/3)}^n:n\in {\mathbb{N}}_0\}$. Because S is closed under multiplication, it is an information semialgebra, which happens to be reduced (see Lemma 3.9). It follows from [23, Theorem 6.2] that $(S,+)$ is an atomic monoid with set of atoms A. For every $n\in \mathbb{N}$, the identity $3=2{\left(\frac{2}{3}\right)}^n+{\sum }_{i=0}^n{\left(\frac{2}{3}\right)}^i$ holds. This implies not only that $(S,+)$ is not a BFM, but also that

  $$M_3:=\left(\begin{array}{ccc}\hfill 1\hfill & \hfill 0\hfill & \hfill 3\hfill \\ \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill \end{array}\right)={\left(\begin{array}{ccc}\hfill 1\hfill & \hfill 0\hfill & \hfill {\left(\frac{2}{3}\right)}^n\hfill \\ \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill \end{array}\right)}^2\left(\begin{array}{ccc}\hfill 1\hfill & \hfill 0\hfill & \hfill 1\hfill \\ \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill \end{array}\right)\left(\begin{array}{ccc}\hfill 1\hfill & \hfill 0\hfill & \hfill \frac{2}{3}\hfill \\ \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill \end{array}\right)\cdots \left(\begin{array}{ccc}\hfill 1\hfill & \hfill 0\hfill & \hfill {\left(\frac{2}{3}\right)}^n\hfill \\ \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill \end{array}\right)$$

  for every $n\in \mathbb{N}$. The matrices on the rightmost part of the above matrix identity are atoms of $T_3{(S)}^{•}$ by [3, Theorem 2.1]. This implies that the set consisting of all the lengths of factorizations of $M_3$ in $T_3{(S)}^{•}$ has infinite size. Hence $T_3{(S)}^{•}$ is not a BFM. This example is an illustration of our Theorem 4.12, which states that the bounded factorization property transfers between $T_n{(S)}^{•}$ and both the additive and the multiplicative monoids of S.

• It follows immediately that the subsemiring $S=\{0,1\}\cup {\mathbb{Q}}_{\ge 2}$ of $\mathbb{Q}$ is a reduced information semialgebra. In addition, one can readily argue that the set of multiplicative atoms of S is $[2,4)\cap \mathbb{Q}$. Then $T_2{(S)}^{•}$ is not an FFM as, for instance, the fact that $a_n=\frac{3n}{n+1}$ and $b_n=\frac{3(n+1)}{n}$ are multiplicative atoms of S for every $n\in \mathbb{N}$ large enough yields infinitely many factorizations of $\left(\begin{array}{cc}\hfill 9\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 1\hfill \end{array}\right)$ in $T_2{(S)}^{•}$, namely, $\left(\begin{array}{cc}\hfill 9\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 1\hfill \end{array}\right)=\left(\begin{array}{cc}\hfill a_n\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 1\hfill \end{array}\right)\left(\begin{array}{cc}\hfill b_n\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 1\hfill \end{array}\right)$. This example illustrates our Theorem 4.17, which states that the finite factorization property transfers between $T_n{(S)}^{•}$ and both the additive and the multiplicative monoids of S.

• One can readily verify that the matrices $\left(\begin{array}{cc}\hfill 1\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 2\hfill \end{array}\right)$ and $\left(\begin{array}{cc}\hfill 1\hfill & \hfill 1\hfill \\ \hfill 0\hfill & \hfill 1\hfill \end{array}\right)$ are atoms of the monoid $T_2{({\mathbb{N}}_0)}^{•}$. As a result, we can see that although both the additive and the multiplicative monoids of the trivial information semialgebra ${\mathbb{N}}_0$ are UFMs, the equality $\left(\begin{array}{cc}\hfill 1\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 2\hfill \end{array}\right){\left(\begin{array}{cc}\hfill 1\hfill & \hfill 1\hfill \\ \hfill 0\hfill & \hfill 1\hfill \end{array}\right)}^2=\left(\begin{array}{cc}\hfill 1\hfill & \hfill 1\hfill \\ \hfill 0\hfill & \hfill 1\hfill \end{array}\right)\left(\begin{array}{cc}\hfill 1\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 2\hfill \end{array}\right)$ shows that $T_2{({\mathbb{N}}_0)}^{•}$ is not an HFM. This example is a particular instance of our Proposition 4.19, which states that $T_n{(S)}^{•}$ fails to be an HFM whenever $n\ge 2$, regardless of the choice of the reduced information semialgebra S.

The present paper focuses on the atomicity of the noncommutative monoid $T_n{(S)}^{•}$ as well as its submonoid $U_n(S)$ consisting of upper triangular matrices whose diagonal entries are multiplicative units of S. Our primary goal here is to establish a more fundamental set of results than those provided in [3]. We characterize when the monoids $T_n{(S)}^{•}$ and $U_n(S)$ are FFMs or BFMs, determine when they satisfy the ACCP, and argue that $T_n{(S)}^{•}$ is almost never an HFM. To do so, we prove that each of these properties, except half-factoriality, transfers back and forth from the monoids $T_n{(S)}^{•}$ and $U_n(S)$ to both the additive and multiplicative monoids of the semiring S. In particular, we give a set of implications analogous to those in Diagram (1.1) but for both $T_n{(S)}^{•}$ and $U_n(S)$.

This paper is structured as follows. In Section 2 we introduce the main objects of study and various related definitions and notations. In Section 3 we introduce the notion of a Puiseux information semialgebra (i.e., an information semialgebra contained in the nonnegative cone of the ordered field $\mathbb{Q}$), and we explore some of the atomic aspects of Puiseux information semialgebras only far enough to use them as our primary source of examples later in Section 4. If S is a reduced information semialgebra and $■\in \{\text{FFM, BFM, ACCP, atomic}\}$, then we say that S is a bi-■ provided that both its additive and multiplicative monoids are ■. Our fundamental results are established in Section 4 and are summarized in the following theorem. Finally, by considering Puiseux information semialgebras, we illustrate that, as in the case of commutative monoids, none of the horizontal implications in the diagram of the following theorem is reversible in general.

Main Theorem

Let S be a reduced information semialgebra. For $n\ge 2$, each implication in the following diagram holds.

Moreover, none of the horizontal implications is, in general, reversible. Finally, $T_n{(S)}^{•}$ is never an HFM when $n\ge 2$.