Lower Separation Axioms for X-top Lattices

We study separation axioms for $X$-top-lattices (i.e. lattices $L$ for which a given subset $X\subseteq L\backslash \{1\}$ admits a \emph{Zariski-like topology}). Such spaces are $T_{0}$ and usually far away from being $T_{2}.$% 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.