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By strict quasigroup I mean a quasigroup with no identity. I've come across one so far in the answer to this question, but I can't seem to find any others. I am particularly interested in finding example of:

  1. Infinite strict quasigroups that are idempotent
  2. Infinite strict quasigroups with neither a left nor a right identity
  3. Infinite strict quasigroups that are idempotent and have neither a left nor a right identity

But other examples "interesting" infinite strict quasigroups are welcome!

MattAllegro
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Nika
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3 Answers3

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At a glance, I think the operation $$x*y=2y-x$$ on $\mathbb{R}$ gives an infinite strict idempotent quasigroup with no left or right identity.

  • In fact, this has a strong "nonidentity" property: if $x\not=y$ then $x*y\not\in\{x,y\}$. This is the strongest failure of the existence of an identity element we can possibly have in an idempotent quasigroup.

This can of course be generalized to $\mathbb{R}^n$ (or indeed a wide class of metric spaces): set $x*y$ to be the unique point $z$ such that $y$ is the midpoint of the line segment $\overline{xz}$.

Noah Schweber
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Another example of quasigroup with infinite underlying set and no identity element is given by the set of the points of a plane cubic curve, with the operation $$a\circ b=c$$ where $a$, $b$ and $c$ are the three intersections of the curve with a same straight line.

No identity: it does not exist any point $e$ suche that $e\circ a=a$ or either $a\circ e=a$ no matter what other point $a$ is.

But, this operation is totally symmetric: if $$a\circ b=c,$$ then $$\sigma(a)\circ\sigma(b)=\sigma(c)$$ where $\sigma$ is any permutation of $(a,b,c)$.

Two references to it:

  1. Etherington, I. (1965). Quasigroups and cubic curves. Proceedings of the Edinburgh Mathematical Society, 14(4), 273-291. doi:10.1017/S001309150000897X
  2. Yu. I. Manin, Cubic Forms, Algebra, Geometry, Arithmetic, Second Edition, North-Holland, 1986
MattAllegro
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    Yes, if I am correct, this is the one that I linked to in my post originally (i.e., this is the only infinite strict quasigroup I had known about before Noah's answer). But I think the way you wrote it was much clearer than in the paper form Etherington, so thank you! By the way, I'm most interested in infinite quasigroups with the underlying set ℕ, so if know of one (or I guess any countably infinite set) I would be Very excited to hear! – Nika Apr 18 '20 at 16:26
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    @Nika oh sorry, I had read this question several times before, and missed it this time. I will definitely update with more examples if I find any! Glad you still read – MattAllegro Apr 18 '20 at 16:30
  • I look forward to any examples you find, is this part of your research? :-) – Nika Apr 18 '20 at 16:35
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    @Nika I'm now an amateur, this was my BSc thesis. Other examples from it for completeness: $(\mathbb{Z},-)$ and $(\mathbb{Q}\backslash{0},/)$. – MattAllegro Apr 18 '20 at 16:38
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The midpoint (or mean) between two points in $\mathbb{R}^n$.

$(a+b)/2=b$ iff $a=b$.

Integrand
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user3229650
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