Open Set Lattices of Subspaces of Spectrum Spaces

Demonstratio mathematica

article

Y.T. Nai¹  D. Zhao¹ 

[1] Mathematics and Mathematics Education National Institute of Education, Nanyang Technological University

关键词: prime element;    S-semiprime element;    m-semiprime element;    hull kernel topology;    pure element;    mp-multiplicative lattice;   

DOI  :  10.1515/dema-2015-0044

学科分类：外科医学

来源: De Gruyter

【 摘 要 】

We take a unified approach to study the open set lattices of various subspaces of the spectrum of a multiplicative lattice L. The main aim is to establish the order isomorphism between the open set lattice of the respective subspace and a sub-poset of L. The motivating result is the well known fact that the topology of the spectrum of a commutative ring R with identity is isomorphic to the lattice of all radical ideals of R. The main results are as follows: (i) for a given nonempty set S of prime elements of a multiplicative lattice L, we define the S-semiprime elements and prove that the open set lattice of the subspace S of Spec(L) is isomorphic to the lattice of all S-semiprime elements of L; (ii) if L is a continuous lattice, then the open set lattice of the prime spectrum of L is isomorphic to the lattice of all m-semiprime elements of L; (iii) we define the pure elements, a generalization of the notion of pure ideals in a multiplicative lattice and prove that for certain types of multiplicative lattices, the sub-poset of pure elements of L is isomorphic to the open set lattice of the subspace Max(L) consisting of all maximal elements of L.

【 授权许可】

CC BY