Second classical Zariski topology on second spectrum of lattice modules
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Let $M$ be a lattice module over a $C$-lattice $L$. Let $Spec^{s}(M)$ be the collection of all second elements of $M$. In this paper, we consider a topology on $Spec^{s}(M)$, called the second classical Zariski topology as a generalization of concepts in modules and investigate the interplay between the algebraic properties of a lattice module $M$ and the topological properties of $Spec^{s}(M)$. We investigate this topological space from the point of view of spectral spaces. We show that $Spec^{s}(M)$ is always $T_{0}-$space and each finite irreducible closed subset of $Spec^{s}(M)$ has a generic point.
Abstract
Let $M$ be a lattice module over a $C$-lattice $L$. Let $Spec^{s}(M)$ be the collection of all second elements of $M$. In this paper, we consider a topology on $Spec^{s}(M)$, called the second classical Zariski topology as a generalization of concepts in modules and investigate the interplay between the algebraic properties of a lattice module $M$ and the topological properties of $Spec^{s}(M)$. We investigate this topological space from the point of view of spectral spaces. We show that $Spec^{s}(M)$ is always $T_{0}-$space and each finite irreducible closed subset of $Spec^{s}(M)$ has a generic point.
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