Open set lattices of subspaces of spectrum spaces

• Autorzy: Nai Y. T. , Zhao D.
• Konferencja: Proceedings of the AAA88 - 88th Workshop on General Algebra Editors for the Special Issue: Anna Romanowska, Jonathan D. H. Smith
• Języki publikacji: EN

Abstrakty

EN

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 M ax(L) consisting of all maximal elements of L.

Słowa kluczowe

EN

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

PL

element pierwszy; topologia; krata; zbiór otwarty

• Wydawca: De Gruyter
• Czasopismo: Demonstratio Mathematica
• Rocznik: 2015
• Tom: Vol. 48, nr 4
• Strony: 637--652
• Opis fizyczny: Bibliogr. 15 poz.

Twórcy

autor

Nai Y. T.

• Mathematics and Mathematics Education, National Institute of Education, Nanyang Technological University, 1 Nanyang Walk, Singapore 637616

autor

Zhao D.

• Mathematics and Mathematics Education, National Institute of Education, Nanyang Technological University, 1 Nanyang Walk, Singapore 637616

Bibliografia

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