Lower Separation Axioms for X-top Lattices

2025/10/25 by Abuhlail, J., Alfaraj, A.
#16D10 #16Y60 #2010: Primary 06A15 #6B30 #FOS: Mathematics #General Topology (math.GN) #Secondary: 13C13

paper · doi:10.48550/arxiv.2510.22355

Abstract

We study separation axioms for X-top-lattices (i.e. lattices L for which a given subset X⊆ L\backslash \1\ admits a Zariski-like topology). Such spaces are T₀ and usually far away from being T₂.% We give graphical characterizations for an X-top-lattice to be T₁, % T_(1)/(4), T_(1)/(2), T_(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.

Citations

• Zariski-like Topologies for Lattices with Applications to Modules over Commutative Rings
• Zariski Topologies for Coprime and Second Submodules
• A Dual Zariski Topology for Modules
• A Zariski Topology for Modules

Related

• Zariski-like Topologies for Lattices with Applications to Modules over Commutative Rings
• Lower spaces of multiplicative lattices
• The structure of decomposable lattices determined by their prime ideals
• Prime ideals in decomposable lattices
• Top(X) within \px ]When lattices meet topology: Top(X) within \px.