Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: 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 $M$).

14
The categories of this paper are: math-ph
16
The categories of this paper are: stat
26
The categories of this paper are: bayes-an
6
The categories of this paper are: astro-ph
18
The categories of this paper are: econ
20
The categories of this paper are: acc-phys
12
The categories of this paper are: nucl-th
34
The categories of this paper are: chem-ph
22
The categories of this paper are: chao-dyn
24
The categories of this paper are: dg-ga
37
The categories of this paper are: supr-con
15
The categories of this paper are: nucl-ex
28
The categories of this paper are: alg-geom
0
The categories of this paper are: hep-ph
10
The categories of this paper are: q-bio
35
The categories of this paper are: mtrl-th