Lower Separation Axioms for X-top Lattices

We study separation axioms for $X$-top-lattices (i.e. lattices $L$ 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 give graphical characterizations for an $X$-top-lattice to be $T_1,$ $$ $T_{\frac{1}{2}},$ $T_{\frac{3}{4}}$ and provide several families of examples/counterexamples that illustrate our results. We apply our results mainly to the prime (resp. maximal, minimal) spectra of prime (resp. maximal, minimal) ideals of commutative (semi)rings.