On an equivalence between integral and involutive residuated structures

Abstract

In the study of residuated lattices (RLs) it sometimes happens that an involutive RL can be reconstructed from its negative cone (consisting of the elements below the unit) via a socalled twist product construction, and moreover these two transformations – restricting to the negative cone and taking a twist product – extend to a categorical equivalence between a variety of involutive RLs and a variety of integral RLs. This enables us to reduce questions about involutive RLs to questions about integral RLs, whose structure is often better understood.