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 $T_{ 1},$ $% T_{ \frac{ 1}{ 4}},$ $T_{ \frac{ 1}{ 2}},$ $T_{ \frac{ 3}{ 4}}$ 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.
Categories: math.GN