Residuated Structures and Orthomodular Lattices

Abstract The variety of (pointed) residuated lattices includes a vast proportion the classes algebras that are relevant for algebraic logic, e.g., $$\ell $$ ℓ -groups, Heyting algebras, MV-algebras, or De Morgan monoids. Among outliers, one counts orthomodular and other varieties quantum algebras. We suggest common framework—pointed left-residuated -groupoids—where structures can all be accommodated. investigate lattice subvarieties pointed -groupoids, their ideals, develop theory left nuclei. Finally, we extend some parts join-completions -groupoids to case, giving new proof MacLaren’s theorem lattices.