  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 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.
