The semilattice congruence \(\mathscr {N}\), which identifies two elements if they generate the same principal filter, plays a significant role in studying the decomposition of semigroups. We investigate the remarkable properties of the semilattice congruence \(\mathscr {N}\) on n-ary semigroups, where \(n\ge 3\), and use these properties to describe the structure of n-ary semigroups which are decomposable into i-simple and regular components for all \(1<i<n\). In particular, we show that each n-ary semigroup which is both regular and intra-regular is decomposable into a semilattice of i-simple and regular n-ary semigroups, and the reverse assertion also holds. Moreover, we prove that an n-ary semigroup is intra-regular if and only if it is a semilattice of i-simple n-ary semigroups. Finally, we discuss the connection between semilattices of i-simple (and regular) n-ary semigroups and semilattices of simple (and regular) n-ary semigroups.

半格同余\(\mathscr {N}\)可以识别两个元素（如果它们生成相同的主滤波器），在研究半群分解中起着重要作用。我们研究了n元半群上的半格同余\(\mathscr {N}\)的显着性质，其中\(n\ge 3\) ，并使用这些性质来描述可分解的n元半群的结构为所有\(1<i<n\)的i -简单和常规组件。特别地，我们证明每个正则和内正则的n元半群都可以分解为i的半格-简单且正则的n元半群，并且相反的断言也成立。此外，我们证明一个n元半群是内正则的当且仅当它是i 个简单n元半群的半格。最后，我们讨论i简单（和正则）n元半群的半格与简单（和正则）n元半群半格之间的联系。