This article investigates derivation theory within the framework of hoop algebras. We introduce and examine f-DERIVATIONS and (f, g)-DERIVATIONS, exploring their properties and providing illustrative examples. We demonstrate that, under specific  conditions, the sets of these DERIVATIONS form bounded distributive lattices and bounded ∨-semilattices. Furthermore, we  investigate s-DERIVATIONS, where s is a square root on the hoop algebra. Notably, we prove that under certain conditions,  the set of fixed points of the s-derivation coincides with the set of idempotents of the hoop algebra. This work  contributes to the understanding of derivation structures in hoop algebras, particularly focusing on the interplay between DERIVATIONS, square roots, and idempotent elements.

Let A be a Banach algebra. Let  d, e be linear mappings on A. First we demonstrate a theorem concerning the continuity of double DERIVATIONS; especially that all of (d, e) -double DERIVATIONS are continuous on semi-simple Banach algebras, in certain case. Afterwards we define a new vocabulary called “(d, e) - higher double derivation" and present a relation between this subject and DERIVATIONS and finally give some theorems and corollaries about the automatic continuity of this notion.

In this paper, we introduce the notions of (ʘ,⊕) -DERIVATIONS and (ϴ,ʘ) -DERIVATIONS for MV -algebras and discuss some related results.We study the connection between these DERIVATIONS on an MV -algebra Aand the DERIVATIONS on its boolean center. We characterize the isotone (ʘ,⊕) -DERIVATIONS and prove that (ϴ,ʘ) -DERIVATIONS are isotone. Finally we determine the relationship between (ʘ,⊕) -derivation and (ϴ,ʘ) -derivation for MV -algebras.

The notions of ξ,-inside GE-derivation and ξ,-outside GE-derivation on a GEalgebra are introduced and its properties are investigated. Conditions for a self-map on GE-algebra to be a ξ,-inside GE-derivation and a ξ,-outside GE-derivation are provided. The ξ,-inside GE-derivation or the ξ,-outside GE-derivation ϱ,are used to form two sets X(ϱ, =ξ, ) and ker(ϱ, ), and GE-subalgebra and GE-, lter are studied for these two sets.

Let A be a Banach algebra. We say that a sequence fDng1 n=0 of continuous operators form A into A is a local higher derivation if to each a 2 A there corresponds a continuous higher derivation fda; ng1 n=0 such that Dn(a) = da; n(a) for each non-negative integer n. We show that if A is a C-algebra then each local higher derivation on A is a higher derivation. We also prove that each local higher derivation on a C-algebra is automatically continuous.

Let A be an algebra. A sequence {dn} of linear mappings on A is called a higher derivation if dn(ab) = Snk=0 dk(a)dn-k(b) for each a, b Î A and each nonnegative integer n. In this paper a notion of an inner higher derivation is given. We characterize all uniformly bounded inner higher DERIVATIONS on Banach algebras and show that each uniformly bounded higher derivation on a Banach algebra A is inner provided that each derivation on A is inner.