An holomorphic study of Smarandache automorphic and cross inverse ploperty loops

TL;DR

This paper investigates the holomorphic structure of certain algebraic loops and quasigroups, establishing conditions under which their holomorphs exhibit Smarandache properties related to automorphic and cross inverse properties.

Contribution

It provides a characterization of when the holomorph of a loop has Smarandache properties based on the triviality of its automorphism group and the loop's own properties.

Findings

Holomorphs of loops are Smarandache AIPL, CIPL, K-loop, Bruck-loop, or Kikkawa-loop under specific conditions.

Triviality of the Smarandache automorphism group is key to the holomorph's properties.

The loop itself must possess certain Smarandache properties for its holomorph to do so.

Abstract

By studying the holomorphic structure of automorphic inverse property quasigroups and loops[AIPQ and (AIPL)] and cross inverse property quasigroups and loops[CIPQ and (CIPL)], it is established that the holomorph of a loop is a Smarandache; AIPL, CIPL, K-loop, Bruck-loop or Kikkawa-loop if and only if its Smarandache automorphism group is trivial and the loop is itself is a Smarandache; AIPL, CIPL, K-loop, Bruck-loop or Kikkawa-loop.