Adjointness and residuation in implicative-ortholattices

Abstract We introduce the mutually adjoint maps on implicative-ortholattices, and we give equivalent conditions for two monotone maps to be mutually adjoint. We also study the adjoint pairs on implicative-ortholattices defined by Sasaki projections, giving a characterization of implicative-orthomodular lattices by adjoint pairs. Moreover, we characterize the implicative-orthomodular lattices by projections in the involution semigroup of their monotone maps having an adjoint. By defining certain Sasaki operations, we prove that an implicative-orthomodular lattice can be transformed into a right residuated po-groupoid. Finally, adding the lattice operations on an implicative-orthomodular lattice, we obtain a weakly divisible and prelinear right residuated lattice.