Residuated Structures and Orthomodular Lattices

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