Abstract We show that the f-vector sets of d-polytopes have non-trivial additive structure: They span affine lattices and are embedded in monoids that we describe explicitly. Moreover, for many large subclasses, such as the simple polytopes, or the simplicial polytopes, there are monoid structures on the set of f-vectors by themselves: “addition of f-vectors minus the f-vector of the d-simplex” always yields a new f-vector. For general 4-polytopes, we show that the modified addition operation does not always produce an f-vector, but that the result is always close to an f-vector. In this sense, the set of f-vectors of all 4-polytopes forms an “approximate affine semigroup”. The proof relies on the fact for d = 4 every d-polytope, or its dual, has a “small facet”. This fails for d > 4. We also describe a two further modified addition operations on f-vectors that can be geometrically realized by glueing corresponding polytopes. The second one of these may yield a semigroup structure on the f-vector set of all 4-polytopes.

中文翻译：

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多胞体 f 向量集上的可加结构

摘要 我们展示了 d-polytope 的 f-vector 集具有非平凡的加法结构：它们跨越仿射格并嵌入我们明确描述的幺半群中。此外，对于许多大的子类，例如简单的多胞体或单纯的多胞体，在 f 向量集合上存在幺半群结构：“f 向量的加法减去 d 单纯形的 f 向量”总是产生一个新的 f 向量。对于一般的 4-polytope，我们表明修改后的加法运算并不总是产生 f 向量，但结果总是接近 f 向量。从这个意义上说，所有 4 个多胞体的 f 向量集形成了一个“近似仿射半群”。证明依赖于 d = 4 的事实，每个 d-polytope 或其对偶都有一个“小面”。这在 d > 4 时失败。我们还描述了对 f 向量的两个进一步修改的加法运算，可以通过粘合相应的多面体在几何上实现。其中的第二个可能会在所有 4-polytopes 的 f-vector 集上产生一个半群结构。