Birkhoff-like sheaf representation for varieties of lattice expansions

Studia Logica 56 (1-2):111 - 131 (1996)
Given a variety we study the existence of a class such that S1 every A can be represented as a global subdirect product with factors in and S2 every non-trivial A is globally indecomposable. We show that the following varieties (and its subvarieties) have a class 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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DOI 10.1007/BF00370143
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References found in this work BETA
Raymond Balbes & Philip Dwinger (1977). Distributive Lattices. Journal of Symbolic Logic 42 (4):587-588.
George Grätzer (1982). Universal Algebra. Studia Logica 41 (4):430-431.

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