The main purpose of this paper is a wide generalization of one of the results abstract algebraic geometry begins with, namely of the fact that the prime spectrum \({\mathrm {Spec}}(R)\) of a unital commutative ring R is always a spectral (= coherent) topological space. In this generalization, which includes several other known ones, the role of ideals of R is played by elements of an abstract complete lattice L equipped with a binary multiplication with \(xy\leqslant x\wedge y\) for all \(x,y\in L\). In fact when no further conditions on L are required, the resulting space can be and is only shown to be sober, and we discuss further conditions sufficient to make it spectral. This discussion involves establishing various comparison theorems on so-called prime, radical, solvable, and locally solvable elements of L; we also make short additional remarks on semiprime elements. We consider categorical and universal-algebraic applications involving general theory of commutators, and an application to ideals in what we call the commutative world. The cases of groups and of non-commutative rings are briefly considered separately.

本文的主要目的是广泛推广抽象代数几何开始的结果之一，即单位交换环R的素数谱\({\mathrm {Spec}}(R)\)是总是一个光谱（=相干）拓扑空间。在这个概括中，包括其他几个已知的，R的理想的作用是由一个抽象的完整格 L 的元素扮演的，该L配备了一个与\(xy\leqslant x\wedge y\)的二进制乘法，对于所有\(x, y\in L\)。事实上，当L没有进一步的条件时是必需的，由此产生的空间可以而且仅被证明是清醒的，我们讨论了足以使其成为光谱的进一步条件。这个讨论涉及建立关于L的所谓素数、激进、可解和局部可解元素的各种比较定理；我们还对半素元素做简短的补充说明。我们考虑涉及交换子一般理论的分类和通用代数应用，以及在我们所谓的交换世界中对理想的应用。群和非交换环的情况分别简要考虑。