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A Class of Multiplicative Lattices

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Abstract

We study the multiplicative lattices L which satisfy the condition a = (a : (a : b))(a : b) for all a, b ∈ 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 ℤ or ℝ. A sharp lattice L localized at its maximal elements are totally ordered sharp lattices. The converse is true if L has finite character.

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Acknowledgement

We thank the referee whose suggestions improved the quality of this paper.

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Authors and Affiliations

  1. Faculty of Mathematics and Computer Science, University of Bucharest, 14 Academiei Str., Bucharest, RO 010014, Romania

    Tiberiu Dumitrescu

  2. Unit 5, Simion Stoilow Institute of Mathematics of the Romanian Academy Research, P.O. Box 1-764, RO-014700, Bucharest, Romania

    Mihai Epure

Authors
  1. Tiberiu Dumitrescu
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  2. Mihai Epure
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Correspondence to Mihai Epure.

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Dumitrescu, T., Epure, M. A Class of Multiplicative Lattices. Czech Math J 71, 591–601 (2021). https://doi.org/10.21136/CMJ.2021.0034-20

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  • DOI: https://doi.org/10.21136/CMJ.2021.0034-20

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