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Stream: learning: reading & references

Topic: Naming of preorder induced by a monoid

Harrison Grodin (May 16 2025 at 21:21):

For a monoid $(M, \cdot, 1)$ , we can define a preorder $x \le y = \exists z.\ x \cdot z = y$ . Does this preorder have a canonical name/reference? I've seen things like "preorder induced by a monoid" (rather long?), "temporal order", "extension order", "prefix order", and such, but it's not clear to me that these are standardized. (Is there even an nLab or Wikipedia page about this ordering?)

Harrison Grodin (May 16 2025 at 21:23):

(Also, $\exists z.\ z \cdot x = y$ works just as well and is distinct when $\cdot$ is not commutative. Maybe this is "suffix order" vs the other is "prefix order"? Or, this is "left-foo" vs the other is "right-foo", where "foo" is a good name for this style of ordering that is used when $\cdot$ is commutative?)

Ralph Sarkis (May 16 2025 at 21:42):

I guess this would be related to Green's relations and the preorders they induce?

Nathan Corbyn (May 17 2025 at 06:59):

I’ve usually seen these called the (left / right)-natural orders

Nathan Corbyn (May 17 2025 at 07:01):

If you see your monoid as a 1-object category—i.e., deloop it—these orders arise as the preorder reflections of the slice and coslice at the single object, respectively