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Abstract <p>The diamond is a five-element lattice which is the smallest non-distributive modular lattice. We establish the dual equivalence of the category with objects being bi-algebraic lattices belonging to the variety generated by the diamond and with morphisms being complete lattice homomorphisms and the category with objects being ordered spaces endowed with an equivalence relation and with morphisms preserving the structure of these spaces. A similar result for the pentagon, a five-element lattice which is the smallest non-modular lattice, was established earlier by W. Dziobiak and the second author. The dual spaces here and there are completely different. However, morphisms are, in each case, an application of an old concept—the minimal join cover refinement property for join covers of an element in a lattice.</p>

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A Duality for the Variety \(\boldsymbol{{S}{P}(M_{3})}\)

  • 期刊论文

  • 发表日期：

    2025年12月1日

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  • A. E. Izyurova,
  • M. V. Schwidefsky

摘要

Abstract

The diamond is a five-element lattice which is the smallest non-distributive modular lattice. We establish the dual equivalence of the category with objects being bi-algebraic lattices belonging to the variety generated by the diamond and with morphisms being complete lattice homomorphisms and the category with objects being ordered spaces endowed with an equivalence relation and with morphisms preserving the structure of these spaces. A similar result for the pentagon, a five-element lattice which is the smallest non-modular lattice, was established earlier by W. Dziobiak and the second author. The dual spaces here and there are completely different. However, morphisms are, in each case, an application of an old concept—the minimal join cover refinement property for join covers of an element in a lattice.