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A squarefree term not occurring in the Leech sequence
arXiv - CS - Discrete Mathematics Pub Date : 2020-03-27 , DOI: arxiv-2003.12213 Benjamin Wells
arXiv - CS - Discrete Mathematics Pub Date : 2020-03-27 , DOI: arxiv-2003.12213 Benjamin Wells
Let \[ \begin{array}{c}\overline{A} = ABCBA\ CBC\ ABCBA,\\ \overline{B} =
BCACB\ ACA\ BCACB,\\ \overline{C} = CABAC\ BAB\ CABAC. \end{array} \] The Leech
sequence $L$ is the squarefree sequence obtained as the limit of the
palindromes \[ A, \overline{A}, \overline{\overline{A}}, \ldots . \] In order
to specify a certain class of pseudorecursive varieties of semigroups, it is
helpful to have a squarefree term in 3 variables such that no substitution
instance occurs as a subterm of $L$. We show that $\kappa_1 = aba\ cbc\ aba\ c$
is such a term. Except for one situation, the doubly-linked term $\kappa_2 =
aba\ cbc\ aba$ will serve, and we focus on it.
中文翻译:
在 Leech 序列中没有出现的 squarefree 项
让 \[ \begin{array}{c}\overline{A} = ABCBA\ CBC\ ABCBA,\\ \overline{B} = BCACB\ ACA\ BCACB,\\ \overline{C} = CABAC\ BAB\ CABAC . \end{array} \] Leech 序列 $L$ 是作为回文极限获得的无平方序列 \[ A, \overline{A}, \overline{\overline{A}}, \ldots 。\] 为了指定某一类半群的伪递归变体,在 3 个变量中有一个无平方项是有帮助的,这样就不会出现替代实例作为 $L$ 的子项。我们证明 $\kappa_1 = aba\ cbc\ aba\ c$ 就是这样一个术语。除了一种情况,双链项 $\kappa_2 = aba\ cbc\ aba$ 将起作用,我们重点关注它。
更新日期:2020-03-30
中文翻译:
在 Leech 序列中没有出现的 squarefree 项
让 \[ \begin{array}{c}\overline{A} = ABCBA\ CBC\ ABCBA,\\ \overline{B} = BCACB\ ACA\ BCACB,\\ \overline{C} = CABAC\ BAB\ CABAC . \end{array} \] Leech 序列 $L$ 是作为回文极限获得的无平方序列 \[ A, \overline{A}, \overline{\overline{A}}, \ldots 。\] 为了指定某一类半群的伪递归变体,在 3 个变量中有一个无平方项是有帮助的,这样就不会出现替代实例作为 $L$ 的子项。我们证明 $\kappa_1 = aba\ cbc\ aba\ c$ 就是这样一个术语。除了一种情况,双链项 $\kappa_2 = aba\ cbc\ aba$ 将起作用,我们重点关注它。