Note on $ p $-ideals set of orthomodular lattices

Ziteng Zhao; Jing Wang; Yali Wu; Ziteng Zhao; Jing Wang; Yali Wu

doi:10.3934/math.20241535

AIMS Mathematics

2024, Volume 9, Issue 11: 31947-31961. doi: 10.3934/math.20241535

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Research article

Note on $ p $-ideals set of orthomodular lattices

1.
School of Mathematics and Science, Hebei GEO University, Shijiazhuang 050031, China
2.
School of Mathematics and Statistics, Qilu University of Technology (Shandong Academy of Sciences), Jinan 250353, China

Received: 16 September 2024 Revised: 28 October 2024 Accepted: 01 November 2024 Published: 11 November 2024
MSC : 03G10, 03G12, 06B10

This paper mainly discusses the problems raised in Kalmbach's book: When are the $ p $-ideals of an irreducible orthomodular lattice well ordered under set inclusion? We give three classes of orthomodular lattices whose $ p $-ideals set is a chain under set inclusion. Furthermore, this article also provides a sufficient and necessary condition for the $ p $-ideals set of orthomodular lattices to be a chain under set inclusion from the perspective of $ L $-algebras, which gives a new point to solve the questions. Moveover, we also give some characterizations about central elements of orthomodular lattice.
- orthomodular lattices,
- $ OM $-$ L $-algebras,
- $ p $-ideals,
- $ L $-ideals,
- well ordered
Citation: Ziteng Zhao, Jing Wang, Yali Wu. Note on $ p $-ideals set of orthomodular lattices[J]. AIMS Mathematics, 2024, 9(11): 31947-31961. doi: 10.3934/math.20241535

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Abstract

This paper mainly discusses the problems raised in Kalmbach's book: When are the $ p $-ideals of an irreducible orthomodular lattice well ordered under set inclusion? We give three classes of orthomodular lattices whose $ p $-ideals set is a chain under set inclusion. Furthermore, this article also provides a sufficient and necessary condition for the $ p $-ideals set of orthomodular lattices to be a chain under set inclusion from the perspective of $ L $-algebras, which gives a new point to solve the questions. Moveover, we also give some characterizations about central elements of orthomodular lattice.

References

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