形式和的半环与 injective 部分变换
Semirings of formal sums and injective partial transformations
AI总结：
本文研究了形式和的半环，探讨了在二元域上injective部分变换的除法问题，提出了对链和环的求解方法。
摘要：
离散动力系统的半环是一种简单的代数模型，用于描述确定性系统中的模ularity。该半环的对象是有限变换（视为有向图并视为同构），两个变换的和对应于在不同的集合上独立应用，而乘积对应于并行应用两者。本文将此半环扩展到包含部分变换；和与积是自然的推广。每个（部分）变换可视为$\mathbb{N}$上的形式和的连接（部分）变换。我们通过在任意半环$\mathbb{S}$上的形式和来推广这一想法。这里考虑$\mathbb{S} = \mathbb{F}_2$的情况，并聚焦于injective部分变换，即链和环的和。虽然在原始离散动力系统的半环中没有已知的高效算法用于求解链的和的除法问题，但本文给出了所有链的和在$\mathbb{F}_2$上的除法问题的解的简洁刻画。然后，我们将这一刻画扩展到任何injective部分变换的除法问题，即在$\mathbb{F}_2$上链和环的和。
英文摘要：
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 $\mathbb{F}_2$.
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