Bipolar Valued Vague Subfields of A Field By Mapping

In this paper, we introduce functions defined on the bipolar-valued vague subfield of a field and investigate their fundamental properties. Since functions play a crucial role in the study of bipolar-valued vague subfields, particular attention is given to the application of homomorphisms within this framework. Furthermore, one or two types of translations are employed in this work, serving as tools to establish communication and relationships between different structures.