Semirings of formal sums and injective partial transformations

Abstract

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 $\mathbb{F}_2$.

Figures (4)

• Figure 1: Sum and product of partial transformations.
• Figure 2: The hexagon lattice $\mathcal{L}$ and the atoms of $V( \mathcal{L} )$.
• Figure 3: Boolean algebra $V( \mathcal{S}^\mathbf{1} )$, with the corresponding parity-check code $P_\mathbf{0}( \mathcal{S}^\mathbf{1} )$ and Boolean algebra $V( \mathcal{S} )$.
• Figure 4: The Boolean algebra $V( \mathcal{L}_5 )$ generated by chains of height $1$, $3$, and $5$.

Theorems & Definitions (74)

• Theorem 2.1
• proof
• Example 2.2
• Remark 2.3
• Example 2.4
• Lemma 2.5
• proof
• Theorem 2.6
• Lemma 2.7
• proof