Editing Quasigroup (section)

== Definitions ==
There are at least two structurally equivalent formal definitions of quasigroup:
* One defines a quasigroup as a set with one [[binary operation]].
* The other, from [[universal algebra]], defines a quasigroup as having three primitive operations.

The [[Homomorphism|homomorphic]] [[Image (mathematics)|image]] of a quasigroup that is defined with a single binary operation, however, need not be a quasigroup, in contrast to a quasigroup as having three primitive operations.{{sfn|ps=|Smith|2007|pp=3,26–27}} We begin with the first definition.

=== Algebra ===
A '''quasigroup''' {{math|(''Q'', ∗)}} is a non-empty [[Set (mathematics)|set]] {{mvar|Q}} with a binary operation {{math|∗}} (that is, a [[magma (algebra)|magma]], indicating that a quasigroup has to satisfy the closure property), obeying the '''Latin square property'''. This states that, for each {{mvar|a}} and {{mvar|b}} in {{mvar|Q}}, there exist unique elements {{mvar|x}} and {{mvar|y}} in {{mvar|Q}} such that both
<math display=block>a \ast x = b</math>
<math display=block>y \ast a = b</math> 
hold. (In other words: Each element of the set occurs exactly once in each row and exactly once in each column of the quasigroup's multiplication table, or [[Cayley table]]. This property ensures that the Cayley table of a finite quasigroup, and, in particular, a finite group, is a [[Latin square]].)  The requirement that {{mvar|x}} and {{mvar|y}} be unique can be replaced by the requirement that the magma be [[Cancellation property|cancellative]].{{sfn|ps=|Rubin|Rubin|1985|p=[https://archive.org/details/equivalentsofaxi0000rubi/page/109 109]}}{{efn|For clarity, cancellativity alone is insufficient: the requirement for existence of a solution must be retained.}}

The unique solutions to these equations are written {{math|''x'' {{=}} ''a'' \ ''b''}} and {{math|''y'' {{=}} ''b'' / ''a''}}.  The operations '{{math|\}}' and '{{math|/}}' are called, respectively, [[left division]] and [[right division]].  With regard to the Cayley table, the first equation (left division) means that the {{mvar|b}} entry in the {{mvar|a}} row is in the {{mvar|x}} column while the second equation (right division) means that the {{mvar|b}} entry in the {{mvar|a}} column is in the {{mvar|y}} row.

The [[empty set]] equipped with the [[Function_(mathematics)#Standard_functions|empty binary operation]] satisfies this definition of a quasigroup.  Some authors accept the empty quasigroup but others explicitly exclude it.{{sfn|ps=|Pflugfelder|1990|p=2}}{{sfn|ps=|Bruck|1971|p=1}}

=== 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.