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Inverse property loop

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

Contents

Definition

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

  1. Existence of left and right inverses: There exist bijective maps \lambda,\rho:L \to L such that \lambda(a) * (a * b) = (b * a) * \rho(a) = b \ \forall \ a, b \in L.
  2. 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, 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 |FULL LIST, MORE INFO
Automorphic inverse property loop |FULL LIST, MORE INFO
Moufang loop |FULL LIST, MORE INFO

Weaker properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
Left-inverse property loop
Right-inverse property loop