发表机构
Recognition Physics Institute; Department of Mathematics, Faculty of Science and Mathematics, University of Niš（识别物理研究所; 尼什大学科学与数学学院数学系）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究三维立方体上闭游走的幺半群，证明交换聚合无法确定约化圈词，给出最小长度14的例证，并区分有限与无界内存，提出非交换商与四元数时钟等识别方法。
AI 中文摘要
我们研究三维立方体$Q_3$的$1$-骨架上的基于闭游走幺半群$\mathcal W$。该图的基本群是自由群$F_5$。我们证明，由阿贝尔群中边权之和给出的每个加性一步读取，都通过有向转移计数进行分解，并且我们展示了两个具有相同转移计数但不同约化圈词的闭游走。因此，交换聚合不能确定约化圈词。约化圈词给出了一个恰当的非交换识别商。我们还确定了具有平凡阿贝尔化但非平凡二阶换位子信息的最短闭游走。其在立方体边度量下的最小长度为$14$。我们还获得了有限内存与无界内存之间的分离。一个两状态读取能分离一个有序对，而对于每个$k\geq1$，存在一个显式的阶乘对，任何$k$状态读取都无法分离。无界栈能恢复每条游走上的约化圈词。对于顶点集上的整数值函数，势读取在闭游走上消失，而占据绑定不是矩形的。更一般地，每个基于占据的约束在一类历史上是矩形的，当且仅当占据向量在该类上是常数。最后，一个声明的四分之一转四元数时钟给出了第二个恰当的顺序敏感同余，与约化词商不可比较。交换恢复的障碍以及最小值$14$，在每个超立方体$Q_n$（$n\geq3$）上仍然有效。
英文摘要
We study the monoid $\mathcal W$ of based closed walks on the $1$-skeleton of the three-dimensional cube $Q_3$. The fundamental group of this graph is the free group $F_5$. We prove that every additive one-step reading, given by a sum of edge weights in an abelian group, factors through the directed transition counts, and we exhibit two closed walks with equal transition counts and different reduced loop words. Hence commutative aggregation does not determine the reduced loop word. The reduced loop word gives a proper noncommutative recognition quotient. We also determine the shortest closed walk with trivial abelianization but nontrivial degree-two commutator information. Its minimum length in the cube edge metric is $14$. We also obtain a separation between finite and unbounded memory. A two-state reading separates an order pair, while for every $k\geq1$, there is an explicit factorial pair which no $k$-state reading separates. An unbounded stack recovers the reduced loop word on every walk. For integer-valued functions on the vertex set, potential readings vanish on closed walks, while occupation binding is not rectangular. More generally, every occupation-based constraint is rectangular on a class of histories if and only if the occupation vector is constant on that class. Finally, a declared quarter-turn quaternion clock gives a second proper order-sensitive congruence, incomparable with the reduced-word quotient. The obstruction to commutative recovery, and the minimum $14$, remain valid on every hypercube $Q_n$, $n\geq3$.
