Splittings in varieties of logic

2025/09/15 by Davey, Brian A., Kowalski, Tomasz, Taylor, Christopher J.
#06F99 (Secondary) #08B15 (Primary) 03G10 #FOS: Mathematics #Logic (math.LO)

paper · doi:10.48550/arxiv.2509.11623

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 splittings algebras, and in the varieties of double Heyting algebras, dually pseudocomplemented Heyting algebras and regular double p-algebras the only splitting algebras are the two-element and three-element chains.

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