Implicative BE algebras as orthogonality spaces

Abstract: We obtain an orthogonality space by endowing an implicative involutive BE algebra with an appropriate orthogonality relation; for these spaces, we also study the particular case of implicative-orthomodular lattices. Moreover, we define the Sasaki projections as well as the commutativity relation on these structures, proving that an implicative involutive BE algebra is an implicative-orthomodular lattice if and only if the commutativity relation is symmetric. We also give characterizations of implicative-Boolean algebras, and we show that the center of an implicative-orthomodular lattice is an implicative-Boolean algebra. We prove that an implicative involutive BE algebra is an implicative-orthomodular lattice if and only if it admits a full Sasaki set of projections. Finally, based on Sasaki maps on implicative involutive BE algebras, we introduce the notion of Sasaki spaces, proving that, in the case when a complete implicative involutive BE algebra admits a full Sasaki set of projections, it is a Sasaki space. We also characterize the Dacey spaces defined by implicative involutive BE algebras.