Abstractly constructed prime spectra

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