{"title":"On Weakly S-Prime Elements of Lattices","authors":"S. E. Atani","doi":"10.22342/jims.30.1.1604.89-99","DOIUrl":null,"url":null,"abstract":"Let £ be a bounded distributive lattice and S a join-subset of £. In this paper, we introduce the concept of S-prime elements (resp. weakly S-prime elements) of £. Let p be an element of £ with S ∧p = 0 (i.e. s∧p = 0 for all s ∈ S). We say that p is an S-prime element (resp. a weakly S-prime element) of £ if there is an element s ∈ S such that for all x, y ∈ £ if p ≤ x ∨ y (resp. p ≤ x ∨ y 6= 1), then p ≤ x ∨ s or p ≤ y ∨ s. We extend the notion of S-prime property in commutative rings to S-prime property in lattices.","PeriodicalId":42206,"journal":{"name":"Journal of the Indonesian Mathematical Society","volume":null,"pages":null},"PeriodicalIF":0.3000,"publicationDate":"2024-04-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of the Indonesian Mathematical Society","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.22342/jims.30.1.1604.89-99","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q4","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
Abstract
Let £ be a bounded distributive lattice and S a join-subset of £. In this paper, we introduce the concept of S-prime elements (resp. weakly S-prime elements) of £. Let p be an element of £ with S ∧p = 0 (i.e. s∧p = 0 for all s ∈ S). We say that p is an S-prime element (resp. a weakly S-prime element) of £ if there is an element s ∈ S such that for all x, y ∈ £ if p ≤ x ∨ y (resp. p ≤ x ∨ y 6= 1), then p ≤ x ∨ s or p ≤ y ∨ s. We extend the notion of S-prime property in commutative rings to S-prime property in lattices.
设 £ 是有界分布网格,S 是 £ 的连接子集。在本文中,我们引入了 S-prime 元素(即弱 S-prime 元素)的概念。设 p 是 £ 的元素,且 S ∧p = 0(即对于所有 s∈ S,s∧p = 0)。如果有一个元素 s∈S 使得对于所有 x, y∈ £ 如果 p ≤ x ∨ y (或者 p≤ x ∨ y 6= 1),那么 p ≤ x ∨ s 或者 p ≤ y ∨ s,我们就说 p 是 £ 的 S-prime 元素(或者弱 S-prime 元素)。