on derivations of pseudo l-algebras

In this article, the focus is on the study of derivations on two types of algebraic structures: pseudo l-algebras and pseudo ckl-algebras. for pseudo l-algebras, the notions of left and right derivations are introduced. these derivations are characterized and equivalent characterizations are given. additionally, the concepts of identity and ideal derivations are defined based on the notion of derivations in pseudo l-algebras. it is proven that any identity derivation is also an ideal derivation. however, an example is provided to demonstrate that not all ideal derivations are identity derivations. moreover, it is shown that ideal left derivations in pseudo l-algebras are idempotent. the article also introduces the notion of fixed point sets in pseudo l-algebras and investigates some properties associated with them. moving on to pseudo ckl-algebras, various properties of derivations in these structures are studied. the relationship between pseudo ckl-algebras and pseudo bck-algebras is established, and it is proven that any pseudo ckl-algebra is also a pseudo bck-algebra. conversely, an example is provided to show that not all pseudo bck-algebras are pseudo ckl-algebras. additionally, it is demonstrated that the contractive derivation of a pseudo ckl-algebra is an identity derivation. we introduce the definition of a pre-ideal and also introduce the definition of a non-empty subset i in pseudo l-algebra, which is d-invariant, and prove that every pre-ideal i in pseudo ckl-algebra is d-invariant, where d is a derivation. overall, the article explores derivations in pseudo l-algebras and pseudo ckl-algebras, providing definitions, characterizations, and examples to illustrate various properties and relationships between these algebraic structures.