Analysis of Complex Entities in Algebra B

We present an exploration of algebra B, a recently published unital and non-associative algebra. Unique complex entities emerged from this algebra, distinct from both quaternion and complex number systems, which we termed treons. We defined the treonic number system and established an isomorphism between this system and the real vector space. Our findings revealed that the treonic representation maintained structural integrity under the defined operations. Based on this foundation, we proceed to determine Euler identity for algebra B within the treonic system. By presenting the fundamental definitions and properties of algebra B, we derived a generalized version of Euler identity applicable within this algebra. This formula revealed the emergence of hyperbolic trigonometric entities, extending the applicability of Euler identity beyond traditional complex numbers. Our results provide a theoretical foundation for a deeper understanding of the properties and behaviors of complex entities within this expanded algebraic framework, thus enabling new theoretical developments and practical applications in the realms of advanced mathematics and theoretical physics.