Geodesic
A class of multiplicative lattices
Czechoslovak Mathematical Journal, Tome 71 (2021) no. 2, pp. 591-601
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We study the multiplicative lattices $L$ which satisfy the condition $ a=(a :\nobreak (a: \nobreak b))(a:b) $ for all $a,b\in L$. Call them sharp lattices. We prove that every totally ordered sharp lattice is isomorphic to the ideal lattice of a valuation domain with value group $\mathbb {Z}$ or $\mathbb {R}$. A sharp lattice $L$ localized at its maximal elements are totally ordered sharp lattices. The converse is true if $L$ has finite character.
We study the multiplicative lattices $L$ which satisfy the condition $ a=(a :\nobreak (a: \nobreak b))(a:b) $ for all $a,b\in L$. Call them sharp lattices. We prove that every totally ordered sharp lattice is isomorphic to the ideal lattice of a valuation domain with value group $\mathbb {Z}$ or $\mathbb {R}$. A sharp lattice $L$ localized at its maximal elements are totally ordered sharp lattices. The converse is true if $L$ has finite character.
DOI : 10.21136/CMJ.2021.0034-20
Classification : 06F99, 13A15, 13F05
Keywords: multiplicative lattice; Prüfer lattice; Prüfer integral domain 
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Dumitrescu, Tiberiu; Epure, Mihai. A class of multiplicative lattices. Czechoslovak Mathematical Journal, Tome 71 (2021) no. 2, pp. 591-601. doi: 10.21136/CMJ.2021.0034-20

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