Higher Separation Axioms for $X$-top Lattices Applications to Commutative (Semi)rings

We study several separation axioms for $X$-top-lattices (i.e. a lattice $L$ for which a given subset $X\subseteq L\mathrm{\backslash }\{1\}$ admits a \emph{% Zariski-like topology}). Such spaces are $T_0$ and usually far away from being $T_2\mathrm{.}$ We provide sufficient/necessary conditions for an $X$-top lattice so that $X$ is $T_2,$ \emph{regular} ($T_3$), \emph{completely regula}r ($T_{3\frac{1}{2}}$), \emph{normal}, \emph{completely normal} or \emph{perfectly normal} ($T_6$). We apply our results mainly to the spectrum of prime (resp. maximal, minimal) ideals of a commutative (semi)ring. We illustrate our results with several examples/counterexamples.