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Are all symmetric idempotent Latin squares known?

There is such a square of order $n$ if and only if $n$ is odd. However, is there a classification of all of them?

(The motivation for the question is here, Latin squares and quasigroups being the same thing).

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Maybe these papers will be of interest for you:

J. Dudek, On Steiner quasigroups

Galuszka_A characterization of commutative and associative groupoids. Discusiones Math., Algebra and Stoch. Methods, 15(1995), 121--125.

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