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.
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We wish to thank Lic. Alfredo Guerin and Dr. Daniel Penazzi for helping us with linguistics aspects. We are indebted to the referee for several helpful suggestions. We also wish to thank Professor Mick Adams for providing us with several reprints and useful e-mail information on the subject.
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Gramaglia, H., Vaggione, D. Birkhoff-like sheaf representation for varieties of lattice expansions. Stud Logica 56, 111–131 (1996). https://doi.org/10.1007/BF00370143
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DOI: https://doi.org/10.1007/BF00370143