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.

Abstract In this paper we define two types of implicative derivations on pseudo-BCI algebras, we investigate their properties and we give a characterization of regular implicative derivations of type II. We also define the notion of a $d$-invariant deductive system of a pseudo-BCI algebra $A$ proving that $d$ is a regular derivation of type II if and only if every deductive system on $A$ is $d$-invariant. It is proved that a pseudo-BCI algebra is $p$-semisimple if and only if the only regular derivation of type II is the identity map. Another main result consists of proving that the set of all implicative derivations of a $p$-semisimple pseudo-BCI algebra forms a commutative monoid with respect to function composition. Two types of symmetric derivations on pseudo-BCI algebras are also introduced and it is proved that in the case of $p$-semisimple pseudo-BCI algebras the sets of type II implicative derivations and type II symmetric derivations are equal.