Equivalence of definitions of inverse property loop

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This article gives a proof/explanation of the equivalence of multiple definitions for the term inverse property loop
View a complete list of pages giving proofs of equivalence of definitions

Statement

The following are equivalent for a loop (L,*).

  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.

Related facts

Proof

(1) implies (2)

It suffices to show that \lambda = \rho; then we can take the inverse map to equal that.

Given: A loop (L,*) with a map \lambda bijective maps \lambda,\rho:L \to L such that \lambda(a) * (a * b) = (b * a) * \rho(a) = b \ \forall \ a, b \in L.

To prove: For any x \in L, \lambda(x) = \rho(x).

Proof: Consider the product (\lambda(x) * x) * \rho(x).

Putting a = b = x and using the property of \rho, this simplifies to \lambda(x). On the other hand, we know that \lambda(x) * x = \lambda(x) * (x * e) = e where e is the identity element, so the expression (\lambda(x) * x) * \rho(x) simplifies to \rho(x). Thus, \lambda(x) = \rho(x).

(2) implies (1)

We can take both \lambda and \rho as the inverse map.