Implicative-ortholattices as orthogonality spaces

Abstract

We obtain an orthogonality space by endowing an implicative-ortholattice with a suitable orthogonality relation; for such spaces, we also investigate the particular case of implicative-orthomodular lattices. Moreover, we define the commutativity relation between two elements of an implicative-ortholattice, as well as the Sasaki projections on this structure. Furthermore, we characterize the implicative-orthomodular lattices and implicative-Boolean algebras, showing that the center of an implicative-orthomodular lattice is an implicative-Boolean algebra. We prove that an implicative-ortholattice is an implicative-orthomodular lattice 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 implicative-ortholattice admits a full Sasaki set of projections, it is a Sasaki space. We also provide a characterization of Dacey spaces arising from implicative-ortholattices.