Abstrakt
It is a well-known that each distributive lattice embeds into a Boolean algebra. We extend this embedding to a substructural setting by embedding suitable lattice-ordered algebras generalizing distributive lattices into involutive residuated lattices.
The crucial observation is that the so-called hemidistributive law introduced by Dunn and Hardegree [2] as an algebraic formulation of the cut rule of sequent calculi provides the appropriate setting for the study of Boolean-like complementation in a substructural context. Since involutive residuated lattices form the algebraic semantics of many substructural logics, this will yield further insight into implication-free fragments of substructural logics.