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Given that the terminology “framed” here clashes with its ordinary use, and given that $\mathbb{R}^n$ equipped with such “framing” is then called a “directed space” anyways (here), why not speak of a “direction” or the like, instead of a “framing”?
I started writing “framed-directed” instead of “framed” in a few places where the context isn’t clear, and I hope that improves the situation a bit. “Framed-directed” can then be abbreviated to “framed” if the context is clear. The following points were considered when the terminology was chosen:
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