Fundamental relations on hyper bi-modules and characterizations of complete hyper bi-modules

A hyper bi-module is a commutative hypergroup that is both a left and a right hypermodule, such that the left and the right multiplications are compatible. We define the fundamental relation on an [Formula: see text]-hyper bi-module, where [Formula: see text] and [Formula: see text] are hyperrings and the left and the right multiplications are compatible. Also, we state some conditions that are equivalent to the transitivity of this relation and finally we characterize the complete [Formula: see text]-hyper bi-modules.