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<p>In this paper we present a construction, on the variety of orthomodular lattices, that generalizes this variety to the non orthocomplemented case: the variety of unsharp orthomodular lattices. Interestingly enough, a relevant deal of the algebraic properties of orthomodular lattices are preserved: as any orthomodular lattice is the union of its Boolean blocks, any unsharp orthomodular lattice will be the union of its building blocks, which can be regarded as regular double Stone algebras. Capitalizing on this construction we will see that it can be turned into a full fledged categorical equivalence between orthomodular lattices indexed by a p-filter and unsharp orthomodular lattices.</p>

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Orthomodular and Unsharp Orthomodular Lattices: A Categorical Equivalence

  • 期刊论文

  • 发表日期：

    2025年7月4日

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  • Antonio Ledda,
  • Gandolfo Vergottini

摘要

In this paper we present a construction, on the variety of orthomodular lattices, that generalizes this variety to the non orthocomplemented case: the variety of unsharp orthomodular lattices. Interestingly enough, a relevant deal of the algebraic properties of orthomodular lattices are preserved: as any orthomodular lattice is the union of its Boolean blocks, any unsharp orthomodular lattice will be the union of its building blocks, which can be regarded as regular double Stone algebras. Capitalizing on this construction we will see that it can be turned into a full fledged categorical equivalence between orthomodular lattices indexed by a p-filter and unsharp orthomodular lattices.