Czechoslovak Mathematical Journal, Vol. 71, No. 2, pp. 591-601, 2021
A class of multiplicative lattices
Tiberiu Dumitrescu, Mihai Epure
Received January 28, 2020. Published online March 15, 2021.
Abstract: We study the multiplicative lattices $L$ which satisfy the condition $ a=(a : (a: 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.
Keywords: multiplicative lattice; Prüfer lattice; Prüfer integral domain
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Affiliations: Tiberiu Dumitrescu, Faculty of Mathematics and Computer Science, University of Bucharest, 14 Academiei Str., Bucharest, RO 010014, Romania, e-mail: tiberiu_dumitrescu2003@yahoo.com, tiberiu@fmi.unibuc.ro; Mihai Epure (corresponding author), Simion Stoilow Institute of Mathematics of the Romanian Academy Research, Unit 5, P.O. Box 1-764, RO-014700 Bucharest, Romania, e-mail: epuremihai@yahoo.com, mihai.epure@imar.ro
