We study quadrangular properties of binary relations on a set $X$~--i.e., properties defined on configurations of four elements--~within an agonistic interpretation, where $xRy$ is interpreted as $x$ ``attacks''~$y$. Such relations induce a suitable notion of ``protection,'' and we provide necessary and sufficient conditions for this notion to be consistent. We characterize the balance property in signed frames in terms of a specific quadrangular property, namely collusivity. In this way, we generalize a classical result in balance theory by offering an alternative method for determining whether a network is polarized. That is, one can identify well-formed groups of agents that agree with one another within the same group (a set of allies) while disagreeing with, or attacking, agents outside the group. Furthermore, we extend the balance theorem to non-symmetric relations, thereby relaxing a condition required in standard balance theory. We conclude by giving a modal characterization of collusive frames, together with corresponding rules in a labeled sequent calculus, and we show that previous modal characterizations of balance are derivable within this system.

翻译：研究集合$X$上二元关系的四元性质（即由四个元素构成的配置所定义的性质），并在对抗性解释框架下进行探讨，其中$xRy$被解释为$x$“攻击”$y$。这类关系可诱导出合适的“保护”概念，我们给出了该概念一致性的充分必要条件。通过将平衡性质与特定的四元性质（即合谋性）相关联，我们刻画了符号框架中的平衡性质。由此，我们推广了平衡理论中的经典结论，提供了一种判断网络是否两极化的替代方法：即可以识别出在组内相互认同（同盟集）且对组外成员持反对或攻击态度的良性群体。此外，我们将平衡定理扩展至非对称关系，从而放宽了标准平衡理论所需的条件。最后，我们给出了合谋框架的模态刻画及其在有标签相继式演算中的对应规则，并证明了先前关于平衡的模态刻画在该系统中均可推导得出。