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It seems to me that the diagrams on wiki about group-like structures are not quite right. For example, the following

https://en.wikipedia.org/wiki/Monoid#/media/File:Algebraic_structures_-_magma_to_group.svg

that appears on the wiki page for monoid https://en.wikipedia.org/wiki/Monoid

The problems I see are that

  1. (nonempty) quasigroup + associativity $\Longleftrightarrow$ (nonempty) group, strictly stronger than (nonempty) inverse semigroup;
  2. semigroup + divisibility (in the sense of how we define quasigroup) $\Longleftrightarrow$ group, also strictly stronger than inverse semigroup;
  3. inverse semigroup + identity $\nRightarrow$ group, the former is sometimes called an inverse monoid.

Similar diagrams appear on other group-like structures, such as https://en.wikipedia.org/wiki/Quasigroup

where the labels "associativity" and "divisibility" do not appear in the lower part of the diagram as in the diagram for monoid but somehow imply similar things, and even if the arrows do not mean the said labels, they still do not make sense since an inverse semigroup is not necessarily a quasigroup.

Did I make any mistake or misunderstand the wiki pages?

ALife
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  • I'm failling to understand why you say "quasigroup + associativity $\Longleftrightarrow$ group, strictly stronger than semigroup;", since the diagram points you to inverse semigroups. An inverse semi-group is a set $S$ with an intern associative binary law such that for all $x$, there exists $y$ such that $xyx=x$ and $yxy=y$. Notice that we don't require an identity, so this is not a group. See https://en.wikipedia.org/wiki/Inverse_semigroup#Examples_of_inverse_semigroups . See also the summary table on the same page. – thibo Feb 21 '21 at 12:50
  • @thibo Thanks for pointing out the issue. Just made an edit to fix. – ALife Feb 21 '21 at 15:21
  • Still, I don't understand why you want that "quasigroup + associativity $Longleftrightarrow $ group" since there exists inverse semigroups which are not groups since they lack identity (and that are indeed quasigroups with associativity, or semigroups with invertibility), see https://math.stackexchange.com/questions/868693/example-of-an-inverse-semigroup or https://math.stackexchange.com/questions/3482542/existence-of-a-commutative-inverse-semigroup-with-no-identity-element?rq=1 – thibo Feb 21 '21 at 16:03
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    It's almost, but not quite, true that an associative quasigroup or a semigroup with divisibility is a group. A nonempty associative quasigroup or semigroup with divisibility is a group. – Jeremy Rickard Feb 21 '21 at 17:12
  • @JeremyRickard Sure, but regardless, an inverse semigroup is not necessarily a quasigroup, and therefore there should not have been an arrow from quasigroup to inverse semigroup with whatever label, right? – ALife Feb 21 '21 at 18:28
  • @thibo an arrow with a label seems to me imply that source concept + label is equivalent to the destination concept, and this is not the case for the diagrams. Please also see my reply to Jeremy. – ALife Feb 21 '21 at 18:31
  • @ALife so the is the issue that in a inverse semigroup what's required is pseudo-inverse $\forall a \exists \tilde{a}, a\tilde{a}a=a$ and $\tilde{a}a\tilde{a}=\tilde{a}$ while for quasigroup one wants divisibility ie $\forall a,b \exists x,y, ax=b$ and $ya=b$ (sorry, I'm not very familiar with the notion of quasigroup)? One counter-example of inverse semi-group not quasigroup would be the 2-elements inverse semigroup ${0,a}$ with $a^2=a$ and $0x=x0=0$. – thibo Feb 21 '21 at 19:01
  • @ALife I agree. The diagram would be correct if "inverse semigroup" were changed to "group or empty set". Even if you interpret the arrows as one way implications, the arrow from "inverse semigroup" to "group" is incorrect. – Jeremy Rickard Feb 21 '21 at 19:45

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