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Splittings in varieties of logic

Brian A. Davey

Mathematics and Statistics, La Trobe University, Victoria 3085, Australia

E-mail Address: B.Davey@latrobe.edu.au

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Tomasz Kowalski

Mathematics and Statistics, La Trobe University, Victoria 3085, Australia

E-mail Address: T.Kowalski@latrobe.edu.au

Corresponding author.

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, and

Christopher J. Taylor

Mathematics and Statistics, La Trobe University, Victoria 3085, Australia

E-mail Address: Chris.Taylor@latrobe.edu.au

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https://doi.org/10.1142/S021819672150034XCited by:0 (Source: Crossref)

Abstract

We study splittings or lack of them, in lattices of subvarieties of some logic-related varieties. We present a general lemma, the non-splitting lemma, which when combined with some variety-specific constructions, yields each of our negative results: the variety of commutative integral residuated lattices contains no splitting algebras, and in the varieties of double Heyting algebras, dually pseudocomplemented Heyting algebras and regular double $p$ $p$ -algebras the only splitting algebras are the two-element and three-element chains.

Communicated by K. Kearnes

Keywords:

AMSC: 08B15, 03G10, 06F99

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