In this paper, we study the four-dimensional real algebra of bihyperbolic numbers. Under consideration of the spectral representation of the bihyperbolic numbers, we give a partial order of bihyperbolic numbers which allows us to obtain some relations in the ordered vector space of bihyperbolic numbers. Moreover, we state that the set of bihyperbolic numbers form a real Banach algebra with a new defined norm. We introduce conjugates, three hyperbolic valued moduli, real moduli, and multiplicative inverse of the bihyperbolic numbers. We give the concept of the absolute value of a bihyperbolic number which generalizes that of real numbers. Also, we represent the polar form of invertible bihyperbolic numbers.

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双曲数的代数性质

在本文中，我们研究了双曲数的四维实数代数。考虑到双曲数的频谱表示，我们给出了双曲数的偏序，这使我们能够在双曲数的有序向量空间中获得某些关系。此外，我们指出双曲数集形成具有新定义范数的实Banach代数。我们介绍了共轭，双曲数值的三个双曲值模，实模和双曲数的乘法逆。我们给出了双曲数的绝对值的概念，它推广了实数的绝对值。同样，我们表示可逆双曲数的极性形式。