Extending the structural homomorphism of LCC loops. (English) Zbl 1106.20051

Csörgő, Piroska

Commentat. Math. Univ. Carol. 46, No. 3, 385-389 (2005).

A loop \(Q\) is said to be left conjugacy closed if the set \(A=\{L_x\mid x\in Q\}\) is closed under conjugation. Let \(Q\) be an LCC loop, let \(\mathcal L\) and \(\mathcal R\) be the left and right multiplication groups of \(Q\), respectively, and let \(I(Q)\) be its inner multiplication group. It is well-known that there exists a homomorphism \(\Lambda\colon\mathcal L\to I(Q)\) determined by \(L_x\to R_x^{-1}L_x\). The purpose of this note is to study some extensions of \(\Lambda\) and to prove the uniqueness of these extensions.

Reviewer: Ladislav Bican (Praha)

MSC:

20N05

Loops, quasigroups

Keywords:

left conjugacy closed loops; multiplication groups; inner mapping groups; homomorphisms

Citations:

Zbl 1101.20035