The friend of my enemy is my enemy, the enemy of my enemy is my friend: Axioms for structural balance and bi-polarity

Structural balance is a simple equilibrium model of positive and negative relationships, such as friendship and enmity. Some relational patterns (e.g. friends sharing an enemy) are balanced; others (e.g. enemies sharing a friend) are not. The model has tested well in a variety of applications, from the social psychology of small groups to the politics of international conflict. Several versions are at least implicit in the literature but had not previously been identified. Here I frame each as an axiom system and prove its equivalence to a version of bi-polarity, the idea that interactive subjects can always be partitioned into two opposing sides.