标题： 基数函数的有界范围 显示英文标题

标题： Bounded ranges of cardinal functions

摘要： 设$\mathbf{x}$为一个（非空）正实数序列。 其成就集$\mathcal{\mathbf{x}}$是$\mathbf{x}$元素的所有可能和的集合。 基数函数为$\mathbf{x}$的函数是$f:\mathcal{A}(\mathbf{x}) \to \mathbb{N}\cup\{\omega,\mathfrak{c}\}$，对于每个$x\in\mathbb{A}(\mathbf{x})$，值$f(x)$等于$x$作为$\mathbf{x}$元素之和表示的方式数目。 在本文中我们考虑序列$\mathbf{x}$的可能范围。 我们提出了一些一般性构造以及集合必须满足的几个条件以成为基数函数的范围。 我们特别关注最大元等于$6$的情况。在这种情况下，我们得到了区间填充序列的基数函数的范围的完整特征。 显示英文摘要

摘要： Let $\mathbf{x}$ be a (non-empty) sequence of positive real numbers. Its achievement set $\mathcal{\mathbf{x}}$ is the set of all the possible sums of the elements of $\mathbf{x}$. The cardinal function of $\mathbf{x}$ is the function $f:\mathcal{A}(\mathbf{x}) \to \mathbb{N}\cup\{\omega,\mathfrak{c}\}$ that for every $x\in\mathbb{A}(\mathbf{x})$ the value $f(x)$ is equal to the number of ways $x$ is represented as a sum of elements of $\mathbf{x}$. In this paper we consider possible ranges of cardinal functions of sequences $\mathbf{x}$. We present some general constructions and several criteria that a set has to satisfy in order to be a range of a cardinal function. We put special attention to the case of sets with maximal element equal to $6$. In this case, in particular, we obtained a full characterisation of sets that are ranges of cardinal functions of interval-filling sequences.