论文标题
长度密度
On length densities
论文作者
论文摘要
对于交换性取消性单体$ m $，我们介绍了m $中的nonunit $ x \的长度密度的概念，表示为$ \ mathrm {ld}（x）$，以及整个monoid $ m $，表示为$ \ mathrm {ld}（ld}（m）$。这个不变的与非单元因素化理论中的三个广泛研究的不变性有关，$ l（x）$，$ \ ell（x）$和$ρ（x）$。我们考虑$ \ mathrm {ld}（x）$和$ \ mathrm {ld}（m）$的一些一般属性，并使用数值半群，puiseux monoids和krull monoids提供了各种示例。虽然我们举了一个具有非理性长度密度的单型$ m $的示例，但我们表明，如果$ m $有限地生成，则$ \ m m ythrm {ld}（m）$是理性的，并且有一个nonunit元素$ x \ in m $ in m $ in m $ in $ \ mathrm {ld}（ld}（ld}（m）= \ mathrm = \ mathrm sage lentys $ {ld lends $ {x）（x）（x）众所周知，$ l（x）$，$ \ ell（x）$和$ρ（x）$（表示$ \ overline {l}（x）$，$ \ overline {\ ell}（x）（x）$，以及$ \ \\\\\\\\\\\\\\\\\\\\\\\虽然众所周知，$ l（x）$，$ \ ell（x）$和$ρ（x）$（表示$ \ overline {l}（x）$，$ \ overline {\ ell}（x）（x）$，以及$ \\ OVERLINECE（x）$）的尤其是惊人的结果， $ \ OVERLINE {\ MATHRM {ld}}（x）= \ lim_ {n \ rightarrow \ infty} \ Mathrm {ld}（x^n）$可能不存在。我们还为$ m $提供了一些有限条件，这些条件迫使$ \ overline {\ mathrm {ld}}}（x）$的存在。
For a commutative cancellative monoid $M$, we introduce the notion of the length density of both a nonunit $x\in M$, denoted $\mathrm{LD}(x)$, and the entire monoid $M$, denoted $\mathrm{LD}(M)$. This invariant is related to three widely studied invariants in the theory of non-unit factorizations, $L(x)$, $\ell(x)$, and $ρ(x)$. We consider some general properties of $\mathrm{LD}(x)$ and $\mathrm{LD}(M)$ and give a wide variety of examples using numerical semigroups, Puiseux monoids, and Krull monoids. While we give an example of a monoid $M$ with irrational length density, we show that if $M$ is finitely generated, then $\mathrm{LD}(M)$ is rational and there is a nonunit element $x\in M$ with $\mathrm{LD}(M)=\mathrm{LD}(x)$ (such a monoid is said to have accepted length density). While it is well-known that the much studied asymptotic versions of $L(x)$, $\ell (x)$ and $ρ(x)$ (denoted $\overline{L}(x)$, $\overline{\ell}(x)$, and $\overlineρ (x)$) always exist, we show the somewhat surprising result that $\overline{\mathrm{LD}}(x) = \lim_{n\rightarrow \infty} \mathrm{LD}(x^n)$ may not exist. We also give some finiteness conditions on $M$ that force the existence of $\overline{\mathrm{LD}}(x)$.
