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A new ordering on the set of all sublattices of a lattice

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Názov: A new ordering on the set of all sublattices of a lattice
Autori: Lavanya, S., Parameshwara Bhatta, S.
Informácie o vydavateľovi: Edited by South China Normal University, Guangzhou; South East Asian Mathmatical Society, Hong Kong
Predmety: Pseudocomplemented lattices, Lattice ideals, congruence relations, chains, orderings on the set of all sublattices of a lattice, Structure theory of lattices, join semidistributive lattice
Popis: Let \(L\) be a lattice and \(\text{Sub}(L)\) the set of all the (nonempty) sublattices of \(L\). The corresponding lattice \(\text{Sub} (L) \cup \{\emptyset\}\) ordered by inclusion is extensively studied. The authors consider another order relation on \(\text{Sub}(L): A \leq B \Leftrightarrow\) for every \(a \in A\) there is a \(b \in B\) such that \(a \leq b\) and for every \(b \in B\) there is an \(a \in A\) such that \(b \geq a\). They prove that together with this partial order \(\text{Sub}(L)\) is a lattice iff \(L\) satisfies one of the following two conditions: (i) all subchains of \(L\) are finite; (ii) \(L\) is chain isomorphic to the usual ordered chain of all the negative integers. Further, if \(L\) is a join semidistributive noetherian and artinian lattice then \(\text{Sub}(L)\) is a join semidistributive lattice iff \(L\) does not contain a direct product of a 2-chain with a 3-chain as sublattice.
Druh dokumentu: Article
Popis súboru: application/xml
Prístupová URL adresa: https://zbmath.org/846087
Prístupové číslo: edsair.c2b0b933574d..1ed926010783af8a302d535107415003
Databáza: OpenAIRE
Popis
Abstrakt:Let \(L\) be a lattice and \(\text{Sub}(L)\) the set of all the (nonempty) sublattices of \(L\). The corresponding lattice \(\text{Sub} (L) \cup \{\emptyset\}\) ordered by inclusion is extensively studied. The authors consider another order relation on \(\text{Sub}(L): A \leq B \Leftrightarrow\) for every \(a \in A\) there is a \(b \in B\) such that \(a \leq b\) and for every \(b \in B\) there is an \(a \in A\) such that \(b \geq a\). They prove that together with this partial order \(\text{Sub}(L)\) is a lattice iff \(L\) satisfies one of the following two conditions: (i) all subchains of \(L\) are finite; (ii) \(L\) is chain isomorphic to the usual ordered chain of all the negative integers. Further, if \(L\) is a join semidistributive noetherian and artinian lattice then \(\text{Sub}(L)\) is a join semidistributive lattice iff \(L\) does not contain a direct product of a 2-chain with a 3-chain as sublattice.