Abstractly constructed prime spectra

TL;DR

This paper generalizes the concept of prime spectra from commutative rings to abstract complete lattices, exploring conditions for spectrality and applications in algebraic and categorical contexts.

Contribution

It introduces a broad framework for prime spectra in complete lattices, extending classical algebraic geometry results to more abstract settings.

Findings

The prime spectrum of a complete lattice can be made spectral under certain conditions.

Comparison theorems for prime, radical, solvable, and locally solvable elements are established.

Applications to categorical and universal algebra contexts are discussed.

Abstract

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.