Digital representation of semigroups and groups
Digital representation of semigroups and groups
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DOI：
10.1007/s00233-008-9074-4
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发表时间：
2008-05
期刊：
影响因子：
0.7
通讯作者：
Stefano Ferri;N. Hindman;D. Strauss
中科院分区：
文献类型：
--
作者：
Stefano Ferri;N. Hindman;D. Strauss
半群（S，S）的数字表示是族<Ft>t∈I，其中I是线性序集，每个F是I的有限非空子集，且S的每个元素都可唯一地表示为形式Hxt，其中His是I的有限子集，每个xt ∈ F t，乘积按指数的递增顺序取。（如果S有一个单位元1，则S ∈ S xt=1。）群G的强数字表示是G的一个数字表示，它具有一个附加性质，即对于eacht∈I，对于somext∈ G和somemt>1，其中mt =2，如果x的阶是无限的，而如果x的阶是有限的，则mt是素数，x的阶是mt的幂。我们证明了任何自由半群都有一个数字表示，|Ft| =1，并且每个阿贝尔群都有一个强的数字表示。我们研究了是否所有的群，甚至所有的有限群都有强数字表示的问题，得到了一些部分结果。最后，我们给出了离散群的Stone-Čech紧化和离散半群的弱概周期紧化在代数中的应用。
Adigital representationof a semigroup (S,⋅) is a family <Ft>t∈I, whereIis a linearly ordered set, eachFtis a finite non-empty subset ofSand every element ofSis uniquely representable in the form Πt∈HxtwhereHis a finite subset ofI, eachxt∈Ftand products are taken in increasing order of indices. (IfShas an identity 1, then Πt∈∅xt=1.) Astrong digital representationof a groupGis a digital representation ofGwith the additional property that for eacht∈I,for somext∈Gand somemt>1 in ℕ wheremt=2 if the order ofxtis infinite, while, if the order ofxtis finite, thenmtis a prime and the order ofxtis a power ofmt. We show that any free semigroup has a digital representation with each|Ft|=1 and that each Abelian group has a strong digital representation. We investigate the problem of whether all groups, or even all finite groups have strong digital representations, obtaining several partial results. Finally, we give applications to the algebra of the Stone-Čech compactification of a discrete group and the weakly almost periodic compactification of a discrete semigroup.