## 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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### References:

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