Automorphic inverse property loop

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

Definition

A loop is said to satisfy the automorphic inverse property if the following two conditions are satisfied:

  • It is a inverse property loop: every element in the algebra loop has a well-defined inverse that plays the role of both a left and right inverse. In other words, for every x, there is an element x^{-1} such that x^{-1} * (x * y) = (y * x) * x^{-1} = y for all y in the loop.
  • if x^{-1} denotes the inverse of x, then:

\! (x * y)^{-1} = x^{-1} * y^{-1}

Relation with other properties

Stronger properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
Left Bruck loop

Weaker properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
Inverse property loop