Conditional probability measure in the algebra of bihyperbolic numbers

An important area of modern mathematics is the study of hypercomplex systems and their possible applications, in particular, for the construction of hypercomplex measure theory, probability theory and mathematical statistics. In this paper, we present a generalization of the notion of a conditional probability measure (real-valued conditional probability) in the case when the measure takes on values in the commutative algebra of bihyperbolic numbers, also called hyperbolic quaternions. Such a generalized conditional probability measure defined on a bihyperbolic probability space for two random events $A$ and $B$ is called a bihyperbolic conditional probability measure or a bihyperbolic conditional probability of event $A$, provided that event $B$ has occurred. It is shown that a bihyperbolic conditional probability measure is defined by all the axioms of a bihyperbolic probability measure. In particular, it is proved that for a fixed event $B$, when the bihyperbolic probability of event $B$ is not zero, the bihyperbolic conditional probability of event $A$, provided that event $B$ has occurred is defined by all the axioms of bihyperbolic probability so that it defines a bihyperbolic probability measure on the measurable space $\left( B, \Sigma_B \right)$, where $\Sigma_B$ is a $\sigma$-algebra of sets of the form $A \cap B$ for $A \in \sigma$. For this purpose, we checked the fulfillment of all three conditions for the definition of a bihyperbolic probability for a bihyperbolic conditional probability. The special cases when the bihyperbolic conditional probability takes on values that are divisors of zero of the algebra of bihyperbolic numbers are investigated. The obtained results can be used in further studies of the relevant sections of probability theory and mathematical statistics.