Studia Logica

, Volume 56, Issue 1–2, pp 111–131 | Cite as

Birkhoff-like sheaf representation for varieties of lattice expansions

  • Hector Gramaglia
  • Diego Vaggione
Article

Abstract

Given a variety ν we study the existence of a class ℱ such that S1 every A ε ν can be represented as a global subdirect product with factors in ℱ and S2 every non-trivial A ε ℱ is globally indecomposable. We show that the following varieties (and its subvarieties) have a class ℱ satisfying properties S1 and S2: p-algebras, distributive double p-algebras of a finite range, semisimple varieties of lattice expansions such that the simple members form a universal class (bounded distributive lattices, De Morgan algebras, etc) and arithmetical varieties in which the finitely subdirectly irreducible algebras form a universal class (f-rings, vector groups, Wajsberg algebras, discriminator varieties, Heyting algebras, etc). As an application we obtain results analogous to that of Nachbin saying that if every chain of prime filters of a bounded distributive lattice has at most length 1, then the lattice is Boolean.

Key words

global subdirect product sheaf lattice ordered algebra Nachbin's theorem Chinese remainder theorem Priestley duality 

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Copyright information

© Kluwer Academic Publishers 1996

Authors and Affiliations

  • Hector Gramaglia
    • 1
  • Diego Vaggione
    • 1
  1. 1.Facultad de Matemática, Astronomía y Física (Fa.M.A.F.)Universidad Nacional de Córdoba, Ciudad UniversitariaCórdobaArgentina

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