Abstract algebra MOC

Quasigroup

A quasigroup is a group-like structure that need not be associative. Formally, a quasigroup (𝑄, ∗) is a magma such that for any 𝑎,𝑏 ∈𝑄 there exist unique 𝑎,𝑏 ∈𝑄 such that algebra

𝑎∗𝑥=𝑏𝑦∗𝑎=𝑏

called the Latin square property since it requires that the resulting Cayley table form a Latin square. Equivalently, we may characterize a quasigroup (𝑄, ∗,/,\) as having left and right division satisfying the following identities (i.e. for all 𝑥,𝑦 ∈𝑄)

𝑦=(𝑦/𝑥)∗𝑥𝑦=(𝑦∗𝑥)/𝑥𝑦=𝑥∗(𝑥\𝑦)𝑦=𝑥\(𝑥∗𝑦)

Proof of equivalence

proof

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