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1. • CommentRowNumber1.
   • CommentAuthorUrs
   • CommentTimeNov 15th 2020
   • (edited Nov 15th 2020)
   • PermaLink
   Author: Urs
   Format: MarkdownItexIn reaction to discussion in [this thread](https://nforum.ncatlab.org/discussion/11876/what-is-the-actual-groupoid-cardinality-of-the-2d-hypercomplex-number-systems/?Focus=87703#Comment_87703): I have changed the relevant text in the entry now to the following: > Notice that the [[complex numbers]] have 2 distinct [[continuous function|continuous]] [[automorphisms]], given, over $\mathbb{R}$, by $e \mapsto \pm e$ (see at _[[automorphism of the complex numbers]]_). > On the other hand, the [[ring of dual numbers]] has a continuous automorphism $e \mapsto k e$ for each $k \in \mathbb{R} \setminus \{0\}$. But the latter space is [[homotopy equivalence|homotopy equivalent]] to two points, corresponding to the [[connected components]] of, again, $e \mapsto \pm e$. > Maybe $e \mapsto \pm e$ are also the two non-trivial continuous automorphisms of the [[perplex numbers]]. > If that is the case, one could say that the [[Euler characteristic]] of the space of continuous automorphisms of each 2d hypercomplex number systems is 2. And maybe with a bit of handwaving towards _[[groupoid cardinality]]_ one might summarize this in saying that there are 3/2 2d hypercomplex number systems. Here I have added the qualifier "continuous" in order to rule out wild automorphisms of the complex numbers. This is not necessary if we demand the automorphism to be over $\mathbb{R}$, but maybe we don't want to demand that here. And given our experience here, I added some verbal scare quotes to the statement that the perplex numbers have precisely two (continuous?) automorphisms. Probably this is unnecessary, but I haven't thought about it. If anyone has, please be invited to edit the entry and strengthen the statement. <a href="https://ncatlab.org/nlab/revision/diff/hypercomplex+number/7">diff</a>, <a href="https://ncatlab.org/nlab/revision/hypercomplex+number/7">v7</a>, <a href="https://ncatlab.org/nlab/show/hypercomplex+number">current</a>

   In reaction to discussion in this thread:

   I have changed the relevant text in the entry now to the following:

   Notice that the complex numbers have 2 distinct continuous automorphisms, given, over $\mathbb{R}$, by $e \mapsto \pm e$ (see at automorphism of the complex numbers).

   On the other hand, the ring of dual numbers has a continuous automorphism $e \mapsto k e$ for each $k \in \mathbb{R} \setminus \{0\}$. But the latter space is homotopy equivalent to two points, corresponding to the connected components of, again, $e \mapsto \pm e$.

   Maybe $e \mapsto \pm e$ are also the two non-trivial continuous automorphisms of the perplex numbers.

   If that is the case, one could say that the Euler characteristic of the space of continuous automorphisms of each 2d hypercomplex number systems is 2. And maybe with a bit of handwaving towards groupoid cardinality one might summarize this in saying that there are 3/2 2d hypercomplex number systems.

   Here I have added the qualifier “continuous” in order to rule out wild automorphisms of the complex numbers. This is not necessary if we demand the automorphism to be over $\mathbb{R}$, but maybe we don’t want to demand that here.

   And given our experience here, I added some verbal scare quotes to the statement that the perplex numbers have precisely two (continuous?) automorphisms. Probably this is unnecessary, but I haven’t thought about it. If anyone has, please be invited to edit the entry and strengthen the statement.

   diff, v7, current

2. • CommentRowNumber2.
   • CommentAuthorUrs
   • CommentTimeNov 15th 2020
   • PermaLink
   Author: Urs
   Format: MarkdownItexAdded the argument that the perplex numbers have two automorphisms (from the discussion [here](https://nforum.ncatlab.org/discussion/11876/what-is-the-actual-groupoid-cardinality-of-the-2d-hypercomplex-number-systems/?Focus=87746#Comment_87746)). Adjusted the following sentence accordingly (and fixed the grammar of the next one). <a href="https://ncatlab.org/nlab/revision/diff/hypercomplex+number/9">diff</a>, <a href="https://ncatlab.org/nlab/revision/hypercomplex+number/9">v9</a>, <a href="https://ncatlab.org/nlab/show/hypercomplex+number">current</a>

   Added the argument that the perplex numbers have two automorphisms (from the discussion here).

   Adjusted the following sentence accordingly (and fixed the grammar of the next one).

   diff, v9, current