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A Simple Way of Getting Large Examples of Osborn Loops
Muhammad Shah
TL;DR
Generating large proper Osborn loops is a computational challenge, despite earlier small finite examples. The paper uses direct products of CC and Moufang loops with boundary constraints, and reports explicit proper examples up to order 2025. Its computational construction still lacks a formal manual proof.
Problem
Earlier finite non-associative Osborn-loop constructions cited in the paper were small, while generating large proper examples remained a computational challenge.
Source: §1 · p. 2Method
Results
Five explicit constructions yield proper Osborn loops up to order 2025.
Source: §3 · pp. 4, 6Takeaways & Limitations
GAP computations establish stable proper Osborn loops from asymmetric non-associative CC and non-associative Moufang factors under the identified boundary metrics.
Source: §5 · p. 7A formal manual proof explaining why these factor selections block collapse into parent subvarieties remains open.
Source: §5 · pp. 7–8
Abstract
from arXiv · showhide
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}.
1 Introduction and Theoretical Framework
The paper presents large proper Osborn-loop construction as a way to study structural boundaries in loop theory. It uses direct products of conjugacy closed and Moufang loops, with constraints to prevent the products from collapsing into either subclass.
1 Introduction and Theoretical Framework
- Examples and counterexamples help test algebraic formulations and delineate the boundaries of open propositions.The paper frames its construction as a way to generate large proper Osborn loops.Source: §1 · pp. 1–2
- Direct products of conjugacy closed and Moufang loops yield Osborn loops, but boundary conditions are needed to ensure they are proper.Here, proper means strictly non-CC and strictly non-Moufang.Source: §1 · p. 2
- Earlier finite non-associative Osborn-loop constructions cited in the paper had orders 16, 24, 36, 48, and 72.Source: §1 · p. 2
- Kinyon’s survey raised open questions about whether every Osborn loop is universal and about proper Osborn varieties with specialized nuclear conditions.Source: §1 · p. 2
2 Direct Product Symmetries and Trivial Collapses
A direct product of a CC loop and a Moufang loop is an Osborn loop, but factor choices can make it collapse into a subvariety. The section identifies associativity and one-sided CC dominance as boundary cases that prevent properness.
2 Direct Product Symmetries and Trivial Collapses
- A CC-loop × Moufang-loop product is Osborn, but properness requires avoiding configurations that collapse into either factor variety.Source: §2 · p. 2
2.1 Case 1: The Associativity Restriction
- An associative CC factor makes the product Moufang, so it cannot meet the paper’s proper-Osborn criterion.The GAP example combines a group of order 7 with a Moufang loop of order 12, yielding a non-associative Moufang loop of order 84 that is Osborn but not proper.Source: §2.1 · pp. 2–3
2.2 Case 2: One-Sided Variety Dominance
- A non-associative CC factor is necessary but insufficient, because Moufang-factor inverse configurations can leave the product CC.Source: §2.2 · p. 3
- Two order-128 products using an order-8 non-associative CC loop and different order-16 Moufang loops remain CC, despite being non-associative Osborn loops.Source: §2.2 · p. 4
3 Synthesis of Large Proper Osborn Loops
Direct products of selected conjugacy closed and Moufang loops yield proper Osborn examples, with constructions reported through order 2025. The examples satisfy the Osborn property while being neither associative, Moufang, nor conjugacy closed.
3 Synthesis of Large Proper Osborn Loops
- Order 96 is obtained from CC loop 8/6 and Moufang loop 12/1, producing a proper Osborn loop.The product is non-associative, neither Moufang nor CC, and Osborn.Source: §3 · pp. 4–5
- Order 160 is obtained from CC loop 8/7 and Moufang loop 20/1, producing a proper Osborn loop.The product is non-associative, neither Moufang nor CC, and Osborn.Source: §3 · p. 5
- Order 192 is obtained from CC loop 8/7 and Moufang loop 24/1, producing a proper Osborn loop.The product is non-associative, neither Moufang nor CC, and Osborn.Source: §3 · p. 5
- Order 300 is obtained from CC loop 25/3 and a Moufang loop of order 12, producing a proper Osborn loop.The product is non-associative, neither Moufang nor CC, and Osborn.Source: §3 · p. 6
- The construction produces proper Osborn loops through order 2025.The order-2025 product is non-associative, not Moufang, not conjugacy closed, and Osborn.Source: §3 · p. 6
4 Library Invariants and Distribution Ledger
The section identifies the GAP loop libraries as the source for selecting factors and records a CC-library invariant for orders 2p.
4 Library Invariants and Distribution Ledger
- GAP DisplayLibraryInfo commands provide the factor-pool boundaries used to deploy the direct-product construction.The section describes a CC loop repository distribution and matching Moufang library indices up to order 243.Source: §4 · p. 6, Table 1 · p. 7, Table 2 · p. 7
- For any odd prime p, the CC library contains exactly one loop structure of order 2p.Source: §4 · p. 7
5 Concluding Remarks and Open Challenges
The paper reports computational support for the direct-product construction but leaves a formal manual proof as an open challenge.
5 Concluding Remarks and Open Challenges
- Computational models using GAP package LOOPS establish the construction under the identified boundary metrics, while a formal manual proof remains open.The open proof challenge is to explain from first algebraic principles why the selected factors block collapse into parent sub-varieties.Source: §5 · pp. 7–8