Structure of the largest idempotent-product free sequences in semigroups
Structure of the largest idempotent-product free sequences in semigroups
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DOI：
10.1016/j.jnt.2018.05.020
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发表时间：
2014-05
影响因子：
0.7
通讯作者：
Guoqing Wang
中科院分区：
文献类型：
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作者：
Guoqing Wang
设S是有限半群，E（S）是S的所有幂等元的集合. Gillam、Hall和威廉姆斯在1972年证明了每个长度至少为T的S值序列|S| −| E（S）|+ 1不是（强）幂等积自由的，在这个意义上，它包含一个非空子序列，其项的乘积，在从序列T导出的顺序中，是幂等的，这肯定了Erdans的问题。他们还表明，|S| −| E（S）|+1是最好的。本文在Gillam，Hall和威廉姆斯等人工作的基础上，确定了长度为|南太平洋（S）|当半群S（不一定是有限的）使得|南太平洋（S）|是有限的，我们引进了一对结构常数的半群，减少到经典的达文波特常数的情况下，有限交换群。
Let S be a finite semigroup, and let E (S) be the set of all idempotents of S. Gillam, Hall and Williams proved in 1972 that every S-valued sequence T of length at least| S|−| E (S)|+ 1 is not (strongly) idempotent-product free, in the sense that it contains a nonempty subsequence the product of whose terms, in the order induced from the sequence T, is an idempotent, which affirmed a question of Erdős. They also showed that the value| S|−| E (S)|+ 1 is best possible. Here, motivated by Gillam, Hall and Williams' work, we determine the structure of the idempotent-product free sequences of length| S∖ E (S)| when the semigroup S (not necessarily finite) is such that| S∖ E (S)| is finite, and we introduce a couple of structural constants for semigroups that reduce to the classical Davenport constant in the case of finite abelian groups.