alternative definition of a quasigroup
In the parent entry, a quasigroup is defined as a set, together with a binary operation on it satisfying two formulas, both of which using existential quantifiers. In this entry, we give an alternative, but equivalent, definition of a quasigroup using only universally quantified formulas. In other words, the class of quasigroups is an equational class.
Definition. A quasigroup is a set with three binary operations (multiplication), (left division), and (right division), such that the following are satisfied:
The two definitions of a quasigroup are equivalent.
Suppose is a quasigroup using the definition given in the parent entry (http://planetmath.org/LoopAndQuasigroup). Define on as follows: for , set where is the unique element such that . Because is unique, is well-defined. Now, let and . Since , and is uniquely determined, this forces . Next, let , then , or . Similarly, define on so that is the unique element such that . The verification of the two right division identities is left for the reader.
Conversely, let be a quasigroup as defined in this entry. For any , let and . Then and . ∎
|Title||alternative definition of a quasigroup|
|Date of creation||2013-03-22 18:28:56|
|Last modified on||2013-03-22 18:28:56|
|Last modified by||CWoo (3771)|