Distributive lattices and some related topologies in comparison with zero-divisor graphs
In this paper,for a distributive lattice $\mathcal L$, we study and compare some lattice theoretic features of $\mathcal L$ and topological properties of the Stone spaces ${\rm Spec}(\mathcal L)$ and ${\rm Max}(\mathcal L)$ with the corresponding graph theoretical aspects of the zero-divisor graph $...
| Main Authors: | , |
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| Format: | Article |
| Language: | English |
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Shahid Beheshti University
2021-01-01
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| Series: | Categories and General Algebraic Structures with Applications |
| Subjects: | |
| Online Access: | https://cgasa.sbu.ac.ir/article_94188_8886cc33fcfa95bf3e8a98c714546517.pdf |
| Summary: | In this paper,for a distributive lattice $\mathcal L$, we study and compare some lattice theoretic features of $\mathcal L$ and topological properties of the Stone spaces ${\rm Spec}(\mathcal L)$ and ${\rm Max}(\mathcal L)$ with the corresponding graph theoretical aspects of the zero-divisor graph $\Gamma(\mathcal L)$.Among other things,we show that the Goldie dimension of $\mathcal L$ is equal to the cellularity of the topological space ${\rm Spec}(\mathcal L)$ which is also equal to the clique number of the zero-divisor graph $\Gamma(\mathcal L)$. Moreover, the domination number of $\Gamma(\mathcal L)$ will be compared with the density and the weight of the topological space ${\rm Spec}(\mathcal L)$. For a $0$-distributive lattice $\mathcal L$, we investigate the compressed subgraph $\Gamma_E(\mathcal L)$ of the zero-divisor graph $\Gamma(\mathcal L)$ and determine some properties of this subgraph in terms of some lattice theoretic objects such as associated prime ideals of $\mathcal L$. |
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| ISSN: | 2345-5853 2345-5861 |