Inverse property loop: Difference between revisions

Revision as of 19:26, 5 March 2010

This article defines a property that can be evaluated for a loop.
View other properties of loops

Definition

An algebra loop $(L, *)$ is termed an inverse property loop or inverse loop or IP-loop if it satisfies the following equivalent conditions:

Existence of left inverses: There exists a bijective map $λ : L \to L$ such that $λ (a) * (a * b) = b \forall a, b \in L$ .
Existence of right inverses: There exists a bijective map $ρ : L \to L$ such that $(a * b) * ρ (b) = a \forall a, b \in L$ .
Existence of two-sided inverses: There exists a bijective map $- 1 : L \to L$ such that $a^{- 1} * (a * b) = (b * a) * a^{- 1} = b$ for all $a, b \in L$ .

Equivalence of definitions

Further information: equivalence of definitions of inverse property loop

Note that for a quasigroup, it is possible to have only the left-inverse property or only the right-inverse property, and even the existence of both left and right inverses does not guarantee the existence of two-sided inverses.

Relation with other properties

Stronger properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
Group

Weaker properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
Left-inverse property loop	the left-inverse map exists
Right-inverse property loop	the right-inverse map exists

@@ Line 3: / Line 3: @@
 ==Definition==
-An [[algebra loop]] <math>(L,*)</math> is termed an '''inverse property loop''' or '''inverse loop''' or '''IP-loop''' if, for there exist bijective maps <math>\lambda</math> and <math>\rho</math> on <math>L</math> such that:
+An [[algebra loop]] <math>(L,*)</math> is termed an '''inverse property loop''' or '''inverse loop''' or '''IP-loop''' if it satisfies the following equivalent conditions:
-<math>\lambda(a) * (a * b) = b \ \forall \ a, b \in L</math>
+# '''Existence of left inverses''': There exists a bijective map <math>\lambda:L \to L</math> such that <math>\lambda(a) * (a * b) = b \ \forall \ a, b \in L</math>.
+# '''Existence of right inverses''': There exists a bijective map <math>\rho:L \to L</math> such that <math>(a * b) * \rho(b) = a \ \forall \ a,b \in L</math>.
+# '''Existence of two-sided inverses''': There exists a bijective map <math>{}^{-1}: L \to L</math> such that <math>a^{-1} * (a * b) = (b * a) * a^{-1} = b</math> for all <math>a,b \in L</math>.
-and:
+===Equivalence of definitions===
-<math>(a * b) * \rho(b) = a \ \forall a,b \in L</math>
+{{further|[[equivalence of definitions of inverse property loop]]}}
-The map <math>\lambda</math> is termed the ''left-inverse map'' and the map <math>\rho</math> is termed the ''right-inverse map''. These maps are unique.
+Note that for a [[quasigroup]], it is possible to have only the left-inverse property or only the right-inverse property, and even the existence of both left and right inverses does not guarantee the existence of two-sided inverses.
 ==Relation with other properties==