Implicative-ortholattices as orthogonality spaces

Abstract We obtain an orthogonality space by endowing an implicative-ortholattice (i-OL) with a suitable orthogonality relation; for such spaces, we also investigate the particular case of implicative-orthomodular lattices (i-OMLs). Moreover, we define the commutativity relation between two elements of an i-OL, as well as the Sasaki projections on this structure. Furthermore, we characterize the i-OMLs and implicative-Boolean algebras (i-Boolean algebras), showing that the center of an i-OML is an i-Boolean algebra. We prove that an i-OL is an i-OML if and only if it admits a full Sasaki set of projections. Finally, based on Sasaki maps on implicative-ortholattices, we introduce the notion of Sasaki spaces, proving that when a complete i-OL admits a full Sasaki set of projections, it is a Sasaki space. We also provide a characterization of Dacey spaces arising from i-OLs.