A BN-algebra is a non-empty set $X$ with a binary operation “$\ast$” and a constant 0 that satisfies the following axioms: $\left(B1\right) x\ast x=0,$ $\left(B2\right) x\ast 0=x,$ and $\left(BN\right) \left(x\ast y\right)\ast z=\left(0\ast z\right)\ast \left(y\ast x\right)$ for all $x, y, z \in X$. A non-empty subset $I$ of $X$ is called an ideal in BN-algebra X if it satisfies $0\in X$ and if $y\in I$ and $x\ast y\in I$, then $x\in I$ for all $x,y\in X$. In this paper, we define several new ideal types in BN-algebras, namely, r-ideal, k-ideal, and m-k-ideal. Furthermore, some of their properties are constructed. Then, the relationships between ideals in BN-algebra with r-ideal, k-ideal, and m-k-ideal properties are investigated. Finally, the concept of r-ideal homomorphisms is discussed in BN-algebra.

This article introduces the concept of a Sheffer stroke BN-algebra by applying the Sheffer stroke operator | to the BN-algebra axioms and aligning it with the axioms of the Sheffer stroke groupoid. From this definition, properties of Sheffer stroke BN algebras are derived, focusing on the relationship between the axioms and the properties of the special element 0. Furthermore, the notions of Sheffer stroke BN-subalgebras, BN-ideals, and BN-homomorphisms are defined, along with normal subsets of Sheffer stroke BN-algebras, and the relationships between these concepts are explored. It is shown that every normal subset in a Sheffer stroke BN-algebra is a Sheffer stroke BN-subalgebra, but the converse is not necessarily true. This implies that every normal BN-ideal in a Sheffer stroke BN-algebra is also a Sheffer stroke BN-subalgebra. Finally, the properties of the kernel of the Sheffer stroke BN-homomorphism are investigated.

The development of modern algebra theory on abstract algebraic structures faces generalization of fundamental concepts, particularly regarding derivations of BG-algebras. A BG-algebra (P;;0) is a non-empty set P with a binary operation * and constant 0 satisfying the following axioms (BG1) pp=0, (BG2) p0=p, and (BG3) (pq)(0q)=p for every p,q ∈ P. Pseudo BG-algebra, which is a generalization of BG-algebra, is an algebra (R;∗,⋄, 0) satisfying the following axioms: (PBG1) p∗0=p⋄0=p, (PBG2) p∗p=p⋄p=0, and (PBG3) (p∗q)⋄(0∗q)= (p⋄q)∗(0⋄q)=p for every p, q ∈ R. In pseudo BG-algebra, the concepts of (l,r)-derivation, (r,l)-derivation, and left derivation have been known as extensions of derivation on BG-algebra. This thesis aims to develop the concept of derivation of pseudo BG-algebra, two main types of derivation namely type 1 derivation and type 2 derivation through an endomorphism-based approach. Pseudo BG-algebra (R;∗,⋄,0) as a mapping to itself, considering the operations ⊛ and ⊙. The formal definitions are (l,r)-f-derivation, (r,l)-f-derivation, and left f-derivation, each with type 1 and type 2. This research develops the concept of (f,g)-derivation on pseudo BG-algebra. The research method uses an axiomatic-deductive approach with theorem proving techniques based on the pseudo BG-algebra axiom system. The overall results of this research demonstrate that the approach using endomorphism, both with a single function f and a pair of functions (f,g) toward derivation on pseudo BG-algebra. The resulting theoretical framework provides a foundation for exploration of derivations on other pseudo algebraic structures and contributes to SDG 4 through the enrichment of theoretical mathematics treasury and SDG 9 through strengthening the mathematical foundation that supports innovation in information technology, cryptography, and secure computing systems that form the backbone of modern era digital infrastructure.