A nucleus on a meet-semilattice A is a closure operation that preserves binary meets. The nuclei form a semilattice \(\mathrm{N }A\) that is isomorphic to the system \({\mathcal {N}}A\) of all nuclear ranges, ordered by dual inclusion. The nuclear ranges are those closure ranges which are total subalgebras (l-ideals). Nuclei have been studied intensively in the case of complete Heyting algebras. We extend, as far as possible, results on nuclei and their ranges to the non-complete setting of implicative semilattices (whose unary meet translations have adjoints). A central tool are so-called r-morphisms, that is, residuated semilattice homomorphisms, and their adjoints, the l-morphisms. Such morphisms transport nuclear ranges and preserve implicativity. Certain completeness properties are necessary and sufficient for the existence of a least nucleus above a prenucleus or of a greatest nucleus below a weak nucleus. As in pointfree topology, of great importance for structural investigations are three specific kinds of l-ideals, called basic open, boolean and basic closed.

相遇半格A上的核是保持二元相遇的闭包操作。原子核形成一个半晶格\(\mathrm{N }A\) ，它与系统\({\mathcal {N}}A\)同构在所有核范围中，按双重包含排序。核范围是那些作为总子代数（l-理想）的闭合范围。在完全 Heyting 代数的情况下，对原子核进行了深入研究。我们尽可能将关于核及其范围的结果扩展到隐含半格的非完全设置（其一元相遇翻译有伴随）。一个核心工具是所谓的 r-态射，即剩余的半晶格同态，以及它们的伴随物，即 l-态射。这种态射传输核范围并保持隐含性。对于在前核之上存在最小核或在弱核之下存在最大核，某些完整性特性是必要和充分的。与无点拓扑一样，对于结构研究非常重要的是三种特定的 l-理想，