The structure of finite commutative idempotent involutive residuated lattices

TL;DR

This paper characterizes finite commutative idempotent involutive residuated lattices as unions of Boolean algebras over a distributive lattice, introduces a new construction called gluing, and explores properties like distributive semilattice fusion reducts and non-local finiteness.

Contribution

It provides a novel structural description and a new construction method for these lattices, enabling the generation of all finite members from Boolean algebras.

Findings

Finite members can be constructed from Boolean algebras using gluing.

The fusion reduct of any finite member is a distributive semilattice.

The variety of these lattices is not locally finite.

Abstract

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.