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We characterize commutative idempotent involutive residuated lattices as disjoint unions of Boolean algebras arranged over a distributive lattice. We use this description to introduce a new construction, called gluing, that allows us to build new members of this variety from other ones. In particular, all finite members can be constructed in this way from Boolean algebras. Finally, we apply our construction to prove that the fusion reduct of any finite member is a distributive semilattice, and to show that this variety is not locally finite.


This is a pre-copy-editing, author-produced PDF of an article accepted for publication in Algebra Universalis, volume 82, in 2021 following peer review. The final publication may differ and is available at Springer via

A free-to-read copy of the final published article is available here.

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