Binary transformation groups and topological fields

The notion of a semitransitive binary action of a group $G$ on a topological space is introduced. A duality theorem is proved, establishing a bijective correspondence between semitransitive distributive binary $G$-spaces and topological fields whose multiplicative group is isomorphic to $G$. This result yields an equivalence between the category of semitransitive distributive binary $G$-spaces and the category of topological fields with multiplicative group $G$. As applications of the duality theorem, two important results are established. It is shown that a finite group can act semitransitively, distributively, and binarily only on finite sets whose cardinality is a power of a prime number. A complete characterization of those groups that can appear as multiplicative groups of topological fields is also obtained.