The Structure Group of Certain K-Loops

Abstract

Let (L,⊕) be a loop, i.e., L is a set with a binary operation ⊕ such that for all a, b ∈ L the equations a ⊕ x = b and y ⊕ a = b have unique solutions x, y ∈ L, and such that there exists an element 0 ∈ L with a ⊕ 0 = 0 ⊕ a = a. The loop L is called a K-loop if the Bol identity

$$ a \oplus (b \oplus (a \oplus c)) = (a \oplus (b \oplus a)) \oplus c $$

holds and the automorphic inverse property

$$ \ominus (a \oplus b) = ( \ominus a) \oplus ( \ominus b) $$

is satisfied, where ⊖a is defined by a⊕ (⊖a) = 0. The Bol identity implies the equality of left and right inverse, i.e., (⊖a) ⊕ a = 0 as well. Thus the phrase “automorphic inverse property” has it’s usual meaning. See [5] for more on loops. In particular, the definition of Bruck-loop in [5, p. 120] is identical with the definition above.