We present syntactic characterisations for the union closed fragments of existential second-order logic and of logics with team semantics. Since union closure is a semantical and undecidable property, the normal form we introduce enables the handling and provides a better understanding of this fragment. We also introduce inclusion-exclusion games that turn out to be precisely the corresponding model-checking games. These games are not only interesting in their own right, but they also are a key factor towards building a bridge between the semantic and syntactic fragments. On the level of logics with team semantics we additionally present restrictions of inclusion-exclusion logic to capture the union closed fragment. Moreover, we define a team based atom that when adding it to first-order logic also precisely captures the union closed fragment of existential second-order logic which answers an open question by Galliani and Hella.

翻译：我们提出工会封闭的第二阶逻辑和逻辑的合成特征,用团队语义来描述。由于工会封闭是一种语义和不可分属性,我们引入的正常形式使得能够处理和更好地理解这一碎片。我们还引入了包容-排斥游戏,这恰恰是相应的模式检查游戏。这些游戏不仅本身很有意思,而且对于在语义和合成语义之间架起桥梁也是一个关键因素。在与团队语义的逻辑层面上,我们还附加了包容-排斥逻辑的限制,以捕捉工会封闭的碎片。此外,我们定义了一个基于原子的团队,在将它添加到第一阶逻辑中时,也精确地捕捉到存在第二阶逻辑的封闭的结合碎片,它回答了加利亚尼和赫拉的开放性问题。