The structure of finite commutative idempotent involutive residuated lattices

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