DERIVED IDENTITIES AND STRUCTURAL PROPERTIES OF OBIC ALGEBRAS

Source abstract

This paper investigates further algebraic properties of obic algebras derived solely from their defining axioms and previously established basic properties. We systematically derive and organize a collection of identities that clarify the behavior of the binary operation. Among the main consequences, we show that the distinguished element 0 is the unique element satisfying $x \cdot 0=x$ for all $x \in X$, establish that $x \cdot y=x$ implies $y=0$, and obtain several zero-producing identities, including $x \cdot(y \cdot(y \cdot x))=0$ and $(x \cdot(x \cdot y)) \cdot y=0$. We also derive rewriting identities that allow certain nested expressions to be transformed into simpler forms. These results provide useful algebraic tools for simplifying computations and proofs and form a convenient basis for further structural investigations of obic algebras.