It is shown that a flat subgroup, H , of the totally disconnected, locally compact group G decomposes into a finite number of subsemigroups on which the scale function is multiplicative. The image, P , of a multiplicative semigroup in the quotient, H/H (1), of H by its uniscalar subgroup has a unique minimal generating set which determines a natural Cayley graph structure on P . For each compact, open subgroup U of G , a graph is defined and it is shown that if P is multiplicative over U then this graph is a regular, rooted, strongly simple P -graph. This extends to higher rank the result of R. Möller that U is tidy for x if and only if a certain graph is a regular, rooted tree.

中文翻译：

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自同构的标度乘法半群的图论描述

结果表明，完全断开的局部紧群 G 的平坦子群 H 分解为有限数量的子半群，在这些子半群上尺度函数是可乘的。H 与其单标子群的商 H/H (1) 中的乘法半群的图像 P 具有唯一的最小生成集，它确定 P 上的自然凯莱图结构。对于 G 的每个紧致开子群 U，定义了一个图，并且证明如果 P 对 U 进行乘法，那么这个图是一个规则的、有根的、强简单的 P -图。这扩展到更高等级的 R. Möller 的结果，即当且仅当某个图是规则的有根树时，U 对 x 是整洁的。