We introduce and discuss a concept of connectedness induced by an n-ary relation (\(n>1\) an integer). In particular, for every integer \(n>1\), we define an n-ary relation \(R_n\) on the digital line \(\mathbb {Z}\) and equip the digital space \(\mathbb {Z}^3\) with the n-ary relation \(R_n^3\) obtained as a special product of three copies of \(R_n\). For \(n=2\), the connectedness induced by \(R_n^3\) coincides with the connectedness given by the Khalimsky topology on \(\mathbb {Z}^3\) and we show that, for every integer \(n>2\), it allows for a digital analog of the Jordan–Brouwer separation theorem for three-dimensional spaces. An advantage of the connectedness induced by \(R_n^3\) (\(n>2\)) over that given by the Khalimsky topology is shown, too.

中文翻译：

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3D数字Jordan-Brouwer分离定理

我们介绍并讨论由n元关系（\（n> 1 \）整数）引起的连通性的概念。特别是，对于每个整数\（n> 1 \），我们在数字行\（\ mathbb {Z} \）上定义n元关系\（R_n \）并配备数字空间\（\ mathbb {Z } ^ 3 \）具有n个三元关系\（R_n ^ 3 \），它是\（R_n \）的三个副本的特殊乘积。对于\（n = 2 \），由\（R_n ^ 3 \）引起的连通性与在\（\ mathbb {Z} ^ 3 \）上的Khalimsky拓扑给出的连通性一致。并且我们表明，对于每个整数\（n> 2 \），它都允许三维空间的Jordan-Brouwer分离定理的数字模拟。还显示了\（R_n ^ 3 \）（\（n> 2 \））所带来的连接性优于Khalimsky拓扑所给出的连接性的优点。