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Zariski-like topologies for lattices with applications to modules over associative rings. (English) Zbl 1430.06005

If \(\mathbb X = \langle X; \tau\rangle\) is a \(T_0\) topological space, and \(K\) is the lattice of closed subsets of \(X\), then the dual lattice \(L=K^{\partial}\) is a complete lattice which has a subset \(\tilde{X}=\{\{p\}\;|\;p\in X\}\subseteq L\) which, when equipped with the Zariski topology, is a space naturally homeomorphic to \(\mathbb X\). That is, the closed subsets of \(\tilde{X}\) in the Zariski topology are the sets \(V(a) = \{\{p\}\in \tilde{X}\;|\;a\leq_L \{p\}\} = \{\{p\}\in \tilde{X}\;|\;p\in a\}\), and the map \(p\mapsto \{p\}\) is a homeomorphism from \(\mathbb X\) to \(\tilde{\mathbb X}\).
This paper starts with an arbitrary complete lattice \(L\) and a proper subset \(X\subsetneq L\) and creates a space on \(X\) with the sets \(V(a) = \{p\in X\;|\;a\leq_L p\}\). \(L\) is called \(X\)-top if the collection of \(V(a)\)’s is closed under union, in which case they form the closed sets of a topology on \(X\). The paper studies the topological properties of the resulting space (separation axioms, connectedness, and compactness), and provides sufficient conditions for the space to be spectral.

MSC:

06B30 Topological lattices
06A15 Galois correspondences, closure operators (in relation to ordered sets)
16D10 General module theory in associative algebras
54B99 Basic constructions in general topology
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