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Theoretical proof demonstrates quasigroup covers for non-quasigroup division groupoids on Cartesian products with regular cardinals, highlighting methods using free quasigroups.
Let G be a division groupoid that is not a quasigroup. For each regular cardinal α>|G| we construct a quasigroup Q on G×α that is a quasigroup cover of G (i.e., G is a homomorphic image of Q and G is not an image of any quasigroup that is a proper factor of Q). We also show how to easily obtain quasigroup covers from free quasigroups.
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Ježek et al. (2024) studied this question.
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