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.

中文翻译：

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一类乘法格

我们研究的乘法格子大号满足条件一个=（一个：（一：b（））一：b为全部）的a，b ∈大号。称他们为锋利的格子。我们证明，每个完全有序的尖晶格都与值域为ℤ或group的估值域的理想晶格同构。定位在其最大元素处的尖晶格L是完全有序的尖晶格。如果L具有有限字符，则相反。