Editing Quasigroup (section)

=== Universal algebra ===
Given some [[algebraic structure]], an [[mathematical identity|identity]] is an equation in which all variables are tacitly [[universal quantifier|universally quantified]], and in which all [[Operation (mathematics)|operations]] are among the primitive operations proper to the structure. Algebraic structures that satisfy axioms that are given solely by identities are called a [[variety (universal algebra)|variety]]. Many standard results in [[universal algebra]] hold only for varieties. Quasigroups form a variety if left and right division are taken as primitive.

A '''right-quasigroup''' {{math|(''Q'', ∗, /)}} is a type {{nowrap|(2, 2)}} algebra that satisfy both identities:
<math display=block>y = (y / x) \ast x</math>
<math display=block>y = (y \ast x) / x</math>

A '''left-quasigroup''' {{math|(''Q'', ∗, \)}} is a type {{nowrap|(2, 2)}} algebra that satisfy both identities:
<math display=block>y = x \ast (x \backslash y)</math>
<math display=block>y = x \backslash (x \ast y)</math>

A '''quasigroup''' {{math|(''Q'', ∗, \, /)}} is a type {{nowrap|(2, 2, 2)}} algebra (i.e., equipped with three binary operations) that satisfy the identities:{{efn|1=There are six identities that these operations satisfy, namely:{{sfn|ps=|Shcherbacov|Pushkashu|Shcherbacov|2021|p=1}}
<math display=block>y = (y / x) \ast x</math>
<math display=block>y = x \backslash (x \ast y)</math>
<math display=block>y = x / (y \backslash x)</math>
<math display=block>y = (y \ast x) / x</math>
<math display=block>y = x \ast (x \backslash y)</math>
<math display=block>y = (x / y) \backslash x</math>
Of these, the first three imply the last three, and vice versa, leading to either set of three identities being sufficient to equationally specify a quasigroup.{{sfn|ps=|Shcherbacov|Pushkashu|Shcherbacov|2021|p=3|loc=Thm.&nbsp;1,&nbsp;2}}
}}
<math display=block>y = (y / x) \ast x</math>
<math display=block>y = (y \ast x) / x</math>
<math display=block>y = x \ast (x \backslash y)</math>
<math display=block>y = x \backslash (x \ast y)</math>

In other words: Multiplication and division in either order, one after the other, on the same side by the same element, have no net effect.

Hence if {{math|(''Q'', ∗)}} is a quasigroup according to the definition of the previous section, then {{math|(''Q'', ∗, \, /)}} is the same quasigroup in the sense of universal algebra. And vice versa: if {{math|(''Q'', ∗, \, /)}} is a quasigroup according to the sense of universal algebra, then {{math|(''Q'', ∗)}} is a quasigroup according to the first definition.