A Boolean network (BN) is a transformation of the set of Boolean configurations of a given length. A trapspace of a BN is a subcube invariant by the BN; a principal trapspace is the smallest trapspace containing a given configuration; a minimal trapspace is one that does not contain any smaller trapspace. In an unrelated development, commutative BNs have been introduced as those networks where all local updates commute. In this paper, we relate those two aspects of BN theory via five main contributions. First, we introduce the trapping graph and the trapping closure of a BN. We also define trapping networks as the networks with transitive general asynchronous graphs and we prove that those are exactly the trapping closures. Second, we show that two BNs have the same collection of (principal) trapspaces if and only if they have the same trapping closure. We then characterise the collections of (principal) trapspaces of BNs. We finally give analogous results for the collections of minimal trapspaces. Third, we prove that commutative networks are trapping, and we classify the collections of principal trapspaces of commutative networks. Fourth, we focus on bijective commutative networks, which we call Marseille networks. We provide several alternative definitions for Marseille networks, and we classify them as special commutative or trapping networks. Fifth, we focus on idempotent commutative networks, which we call Lille networks. We provide several alternative definitions for Lille networks, we classify them as special commutative or trapping networks, and we relate them to globally idempotent networks. Our investigations of Marseille and Lille networks also highlight relations amongst the asynchronous, general asynchronous, and trapping graphs of Boolean networks, as well as the structure of trapping networks in general.

翻译：布尔网络（BN）是对固定长度布尔配置集合的一个变换。布尔网络的陷阱空间是由该BN保持不变的子立方体；主陷阱空间是包含给定配置的最小陷阱空间；最小陷阱空间是不包含任何更小陷阱空间的陷阱空间。在另一独立的研究方向上，交换布尔网络被定义为所有局部更新函数可交换的网络。本文通过五大贡献将布尔网络理论的这两个方面联系起来。首先，我们引入了布尔网络的陷阱图和陷阱闭包。我们还定义了具有传递性一般异步图的陷阱网络，并证明这些网络恰好是陷阱闭包。其次，我们证明两个布尔网络具有相同的（主）陷阱空间集合当且仅当它们具有相同的陷阱闭包，进而刻画了布尔网络（主）陷阱空间集合的特征，并最终给出了最小陷阱空间集合的类似结果。第三，我们证明交换网络是陷阱网络，并分类了交换网络的主陷阱空间集合。第四，我们聚焦于双射交换网络，即马赛网络。我们为马赛网络提供了多种等价定义，并将其归类为特殊的交换网络或陷阱网络。第五，我们聚焦于幂等交换网络，即里尔网络。我们为里尔网络提供了多种等价定义，将其归类为特殊的交换网络或陷阱网络，并揭示了其与全局幂等网络的关联。对马赛网络和里尔网络的研究还凸显了布尔网络的异步图、一般异步图与陷阱图之间的关系，以及陷阱网络的整体结构。