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![[parent]](http://images.planetmath.org:8080/images/uparrow.png)
alternative definition of a quasigroup
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(Definition)
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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 $Q$ with three binary operations $\cdot$ (multiplication), $\backslash$ (left division), and $/$ (right division), such that the following are satisfied:
- $(Q,\cdot)$ is a groupoid (not in the category theoretic sense)
- (left division identities) for all $a,b\in Q$ , $a \backslash (a \cdot b)=b$ and $a\cdot (a \backslash b) = b$
- (right division identities) for all $a,b\in Q$ , $(a \cdot b)/ b=a$ and $(a/b) \cdot b = a$
Proposition 1 The two definitions of a quasigroup are equivalent.
Proof. Suppose $Q$ is a quasigroup using the definition given in the parent entry. Define $\backslash$ on $Q$ as follows: for $a,b\in Q$ , set $a\backslash b:=c$ where $c$ is the unique element such that $a\cdot c = b$ . Because $c$ is unique, $\backslash$ is well-defined. Now, let $x = a\cdot b$ and $y = a\backslash x$ . Since $a\cdot y = x = a \cdot b$ , and $y$ is uniquely determined, this forces $y=b$ . Next, let $x=a\backslash b$ , then $a \cdot x =b$ , or $a \cdot (a\backslash b) = b$ . Similarly, define $/$ on $Q$ so that $a/b$ is the unique element $d$ such that $d\cdot b=a$ . The verification of the two right division identities is left for the reader.
Conversely, let $Q$ be a quasigroup as defined in this entry. For any $a,b\in Q$ , let $c=a\backslash b$ and $d=b/a$ . Then $a \cdot c = a \cdot (a \backslash b) = b$ and $d \cdot a = (b/a) \cdot a = b$ . 
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"alternative definition of a quasigroup" is owned by CWoo.
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Cross-references: conversely, forces, well-defined, element, definitions, identities, category, groupoid, multiplication, equational class, class, words, equivalent, existential quantifiers, formulas, binary operation, quasigroup, parent
There are 2 references to this entry.
This is version 3 of alternative definition of a quasigroup, born on 2008-10-07, modified 2008-10-11.
Object id is 11158, canonical name is AlternativeDefinitionOfAQuasigroup.
Accessed 1104 times total.
Classification:
| AMS MSC: | 20N05 (Group theory and generalizations :: Other generalizations of groups :: Loops, quasigroups) |
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Pending Errata and Addenda
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