一种获取Osborn环大实例的简单方法
A Simple Way of Getting Large Examples of Osborn Loops
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中文总结 AI 辅助
本文提出通过非结合共轭封闭环与Moufang环的直积生成大的真Osborn环，给出边界约束并验证了直至阶2025的实例。
中文摘要 AI 辅助
例子和反例界定了数学命题的边界，检验基础猜想，并解决抽象代数中的开放结构问题。在环理论中，Osborn环是Moufang环的一个重要推广，满足一个特殊的变量夹层恒等式。虽然已知有小的有限非结合例子，但构造大的真Osborn环（既非共轭封闭也非Moufang的环）实例仍然是计算上的瓶颈。在本文中，我们通过取非结合共轭封闭（CC）环与Moufang环的直积，建立了一个生成大的真Osborn环的高效框架。我们概述了防止结构坍缩为子簇所需的边界约束，并提供了直至阶2025的显式例子，这些例子通过GAP包\ exttt{LOOPS}验证。
英文摘要
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}.
发表机构
- Department of Mathematics, Government Degree College Wadpagga, Peshawar, Pakistan（白沙瓦瓦德帕加政府学位学院数学系）
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