论文标题
上层三角矩阵的因素在信息半序列上
Factorizations in upper triangular matrices over information semialgebras
论文作者
论文摘要
如果每个非零单位元素都是不可减少的乘积，则整体域（或交换取消型单体）是原子质的，并且如果每一个主体理想最终都可以稳定，则可以满足ACCP。自1970年代以来，已经研究了这两种属性之间的相互作用。如果每个元素仅具有有限的因数化，并且满足有限的分解属性（BFP），则原子域（或单型）（或单型）可满足有限分解属性（FFP），如果对于每个元素的每个因素中的原子数量都有一个共同的界限，则它的每个因素都有一个共同的界限。自1990年由Anderson，Anderson和Zafrullah引入以来，这两个属性已被系统地研究。Noetherian域满足BFP，而Dedekind域则满足FFP。众所周知，对于交换性取消型单体（尤其是整体域），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~$S$.我们还表明，如果一个由单位三角矩阵组成的submonoid $ u_n（s）$替换$ t_n（s）^\ bullet $，则会发生类似的转移行为。结果，我们发现链条ffp $ \ rightarrow $ bfp $ \ rightarrow $ accp $ \ rightarrow $ atomic也容易组成，包括包括非交通型单sonoids $ t_n（s）^\ bullet $和$ u_n（s）$。最后，我们构建了各种合理信息的半序列，以验证一般而言，确定的含义都不是可逆的。
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.
