The semiring of discrete dynamical systems is a simple algebraic model for modularity in deterministic systems. The objects of the semiring are finite transformations (viewed as directed graphs and regarded up to isomorphism), the sum of two transformations corresponds to applying them independently on distinct sets, and the product corresponds to applying both transformations in parallel. In this paper, we extend this semiring to include partial transformations; the sum and product are natural generalisations. Each (partial) transformation can be viewed as a sum (over $\mathbb{N}$) of connected (partial) transformations. We generalise this idea by working in semirings of formal sums over any semiring $\mathbb{S}$. Here we consider the case where $\mathbb{S} = \mathbb{F}_2$, the binary field, and we focus on injective partial transformations, i.e. sums of chains and cycles. While no efficient algorithm for the division problem for sums of cycles in the original semiring of discrete dynamical systems is known, we give a concise characterisation of all the solutions of the division problem for sums of cycles over $\mathbb{F}_2$. We then extend this characterisation to dividing any injective partial transformations, i.e. sums of chains and cycles over $\mathbb{F}_2$.

翻译：离散动力系统的半环是确定性系统中模块化的一个简单代数模型。该半环的对象是有限变换（视为有向图且在同构意义下考虑），两个变换的加法对应将其独立用于不相交的集合，乘法对应并行执行两个变换。本文将该半环扩展至包含部分变换；加法与乘法为自然推广。每个（部分）变换可视为连通（部分）变换的（$\mathbb{N}$上）和。我们通过在任何半环$\mathbb{S}$上建立半形式幂和的半环来推广这一思想。此处考虑$\mathbb{S} = \mathbb{F}_2$（二元域）的情形，并聚焦于单射部分变换，即链与环的和。尽管原始离散动力系统半环中环的除法问题尚无已知的高效算法，我们给出了$\mathbb{F}_2$上环之除法问题所有解的简洁刻画。进而将此刻画推广至任意单射部分变换（即$\mathbb{F}_2$上链与环的和）的除法。