On left (θ, ϕ)-derivations in BCI-algebras

The notion of (regular) left (θ, ϕ)-derivations of a BCI-algebra is introduced, some useful examples are discussed, and related properties are investigated. Conditions for a left (θ, ϕ)-derivation to be regular are provided. The concepts of a d(θ,ϕ)-invariant left (θ, ϕ)-derivation and θ-ideal are introduced, and their relations are discussed. Furthermore, some more interesting results are established.