A Simple Way of Getting Large Examples of Osborn Loops
Abstract
Examples and counterexamples define the boundaries of mathematical propositions, test foundational conjectures, and resolve open structural problems in abstract algebra. In loop theory, an Osborn loop is a crucial generalization of a Moufang loop satisfying a specialized variable-sandwiching identity. While small finite non-associative examples are known, constructing large examples of proper Osborn loops (loops that are neither conjugacy closed nor Moufang) remains a computational bottleneck. In this paper, we establish an efficient framework for generating large proper Osborn loops by taking the direct product of non-associative conjugacy closed (CC) loops and Moufang loops. We outline the boundary constraints necessary to prevent structural collapse into sub-varieties and provide explicit examples up to order 2025 validated via the GAP package \texttt{LOOPS}.
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BibTeX
@misc{shah2026simple,
title = {{A Simple Way of Getting Large Examples of Osborn Loops}},
author = {Muhammad Shah},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.09176},
eprint = {2609.09176},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.09176},
}
RIS
TY - UNPB
TI - A Simple Way of Getting Large Examples of Osborn Loops
AU - Muhammad Shah
PY - 2026
DA - 2026/09/01
PB - arXiv
DO - 10.48550/arxiv.2609.09176
UR - https://arxiv.org/abs/2609.09176
AB - Examples and counterexamples define the boundaries of mathematical propositions, test foundational conjectures, and resolve open structural problems in abstract algebra. In loop theory, an Osborn loop is a crucial generalization of a Moufang loop satisfying a specialized variable-sandwiching identity. While small finite non-associative examples are known, constructing large examples of proper Osborn loops (loops that are neither conjugacy closed nor Moufang) remains a computational bottleneck. In this paper, we establish an efficient framework for generating large proper Osborn loops by taking the direct product of non-associative conjugacy closed (CC) loops and Moufang loops. We outline the boundary constraints necessary to prevent structural collapse into sub-varieties and provide explicit examples up to order 2025 validated via the GAP package \texttt{LOOPS}.
ER -