Zariski-like Topologies for Lattices with Applications to Modules over Commutative Rings

We study Zariski-like topologies on a proper class $X\varsubsetneqq L$ of a complete lattice $\mathcal{L}=(L,\wedge ,\vee ,0,1)$. We consider $X$ with the so called classical Zariski topology $(X,\tau ^{cl})$ and study its topological properties (e.g. the separation axioms, the connectedness, the compactness) and provide sufficient conditions for it to be $\textit{spectral}$. We say that $\mathcal{L}$ is $X$\emph{-top} iff% \begin{equation*} \tau :=\{X\backslash V(a)\mid a\in L\},\text{ where }V(a)=\{x\in L\mid a\leq x\} \end{equation*}% is a topology. We study the interplay between the \textit{algebraic properties} of an $X$-top complete lattice $\mathcal{L}$ and the $\textit{% topological properties}$ of $(X,\tau ^{cl})=(X,\tau ).$ Our results are applied to several spectra which are proper classes of $\mathcal{L}% :=LAT(_{R}M)$ where $M$ is a left module over an arbitrary associative ring $% R$ (e.g. the spectra of prime, coprime, fully prime submodules) of $M$ as well as to several spectra of the dual complete lattice $\mathcal{L}^{0}$ (e.g. the spectra of first, second and fully coprime submodules of $M$).