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On the minimal number of generators of endomorphism monoids of full shifts
Natural Computing ( IF 1.7 ) Pub Date : 2020-02-14 , DOI: 10.1007/s11047-020-09785-4
Alonso Castillo-Ramirez

For a group G and a finite set A, denote by \(\mathrm{End}(A^G)\) the monoid of all continuous shift commuting self-maps of \(A^G\) and by \(\mathrm{Aut}(A^G)\) its group of units. We study the minimal cardinality of a generating set, known as the rank, of \(\mathrm{End}(A^G)\) and \(\mathrm{Aut}(A^G)\). In the first part, when G is a finite group, we give upper and lower bounds for the rank of \(\mathrm{Aut}(A^G)\) in terms of the number of conjugacy classes of subgroups of G. In the second part, we apply our bounds to show that if G has an infinite descending chain of normal subgroups of finite index, then \(\mathrm{End}(A^G)\) is not finitely generated; such is the case for wide classes of infinite groups, such as infinite residually finite or infinite locally graded groups.



中文翻译:

关于全班态内同态半体生成器的最小数目

为一组G ^和一组有限的,表示由\(\ mathrm {完}(A ^ G)\)的所有连续移位的通勤的自映射的幺\(A ^ g \)和由\(\ mathrm {Aut}(A ^ G)\)其单位组。我们研究了发电机组的最小基数,被称为,的\(\ mathrm {完}(A ^ G)\)\(\ mathrm {AUT}(A ^ G)\) 。在第一部分中,当G ^是一个有限群,我们给出的等级的上限和下限\(\ mathrm {AUT}(A ^ G)\)中的亚组的共轭类的数量方面ģ。在第二部分中,我们应用边界来表明如果G具有有限索引的普通子组的无限下降链,则\(\ mathrm {End}(A ^ G)\)并不是有限生成的;广泛类别的无限组就是这种情况,例如无限残差有限或无限局部分级的组。

更新日期:2020-04-23
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