File talk:Magma to group4.svg
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I think this diagram's incorrect, inheriting an incorrect detail from its predecessor 'Magma to group3.svg'. An inverse semigroup isn't necessarily a quasigroup, as a quasigroup has the full 'Latin square property' and an inverse semigroup only has the weaker property of each element having an inverse. 2601:18C:C900:BB00:BD5A:4EFA:31B5:4CE 02:22, 15 April 2021 (UTC)
- I agree - at least part of the problem is the lack of labelling on the 'interior' arrows, particularly the red ones, suggesting that the difference between an inverse semigroup and a generic semigroup is the same as the difference between a quasigroup and a generic magma - which as you say, it is not. The 'invertibility' on the bottom-right red arrow is equivalent to divisibility because of the existence of an identity, whereas the invertibility of an inverse semigroup is not: as can be seen in the Wikipedia article for inverse semigroup, an i.s. is not only not a quasigroup, but it can have an identity (that is, it can be an inverse monoid) and still not be a group... contrary to what this diagram currently implies. Adam Dent (talk) 07:21, 15 June 2021 (UTC)
- Also, an associative quasigroup -- assuming it is not empty -- is a group. Overall I'd suggest reverting to an earlier version of this diagram, say version 2, and try to make a correct new version. — Preceding unsigned comment was added by 213.236.218.114 (talk) 10:37, 11 August 2021 (UTC)
- See also here: https://en.wikipedia.org/wiki/Template_talk:Group-like_structures#Removed_'Inverse_Semigroup' 213.22.81.218 17:56, 11 November 2022 (UTC)
- Maybe Ethaniel or Tomruen could create one of these puppies with the Unital magma category and the colors but without Inverse semigroup category. One could call it Magma_to_group5 or something like that. Interestingly, I had asked about other pathways way back in 2008, and was told that the only possible magma with the cancellation property (divisibility in this case) and associativity but without an identity element was the empty set, and that doesn't count as a quasigroup as being non-empty is part of the definition, although I guess the empty set is considered a semigroup (it associativity isn't violated, it can apparently be considered to exist). So, it's almost like you could draw a green arrow from Quasigroup to Group (or perhaps be fancy and try to merge that arrows with the green arrow coming from Loop). Or maybe just replace "Inverse semigroup" with "Empty Set" or "Empty semigroup", and if you wanted a complete "cube" you could add "plus Empty set" in small letters below quasigroup, which together would make up the category of all magmas where division was always possible (but that are not necessarily associative or with identity elements or even with any element. Who decided that the definition of quasigroups should exclude empty magmas anyway? Okay, end of rant. Kevin Lamoreau (talk) 00:29, 22 December 2022 (UTC)
- Just replace "inverse semigroup" with "associative quasigroup". That is correct, whereas the current "inverse semigroup" is utterly incorrect. Quondum (talk) 00:32, 25 June 2023 (UTC)
- Maybe Ethaniel or Tomruen could create one of these puppies with the Unital magma category and the colors but without Inverse semigroup category. One could call it Magma_to_group5 or something like that. Interestingly, I had asked about other pathways way back in 2008, and was told that the only possible magma with the cancellation property (divisibility in this case) and associativity but without an identity element was the empty set, and that doesn't count as a quasigroup as being non-empty is part of the definition, although I guess the empty set is considered a semigroup (it associativity isn't violated, it can apparently be considered to exist). So, it's almost like you could draw a green arrow from Quasigroup to Group (or perhaps be fancy and try to merge that arrows with the green arrow coming from Loop). Or maybe just replace "Inverse semigroup" with "Empty Set" or "Empty semigroup", and if you wanted a complete "cube" you could add "plus Empty set" in small letters below quasigroup, which together would make up the category of all magmas where division was always possible (but that are not necessarily associative or with identity elements or even with any element. Who decided that the definition of quasigroups should exclude empty magmas anyway? Okay, end of rant. Kevin Lamoreau (talk) 00:29, 22 December 2022 (UTC)
- See also here: https://en.wikipedia.org/wiki/Template_talk:Group-like_structures#Removed_'Inverse_Semigroup' 213.22.81.218 17:56, 11 November 2022 (UTC)
