Generalizations of Derivations in BCI-Algebras

In the present paper we introduced the notion of (θ ,φ)-derivations of a BCI-algebra X . Some interesting results on inside (or outside) (θ ,φ)-derivations in BCI-algebras are discussed. It is shown that for any commutative BCI-algebra X , every inside (θ ,φ)derivation of X is isotone. Furthermore it is also proved that for any outside (θ ,φ)-derivation d(θ ,φ) of a BCI-algebra X , d(θ ,φ)(x) = θ (x)∧d(θ ,φ)(x) if and only if d(θ ,φ)(0) = 0 for all x ∈ X .