Semirings of formal sums and injective partial transformations

TL;DR

This paper extends the algebraic model of discrete dynamical systems to include partial transformations and formal sums over semirings, providing new characterizations for division problems in this context.

Contribution

It generalizes the semiring of transformations to include partial and formal sums over any semiring, focusing on injective partial transformations over _2, and characterizes solutions to division problems.

Findings

Characterization of solutions for division of sums of cycles over _2.

Extension of division characterization to injective partial transformations.

No known efficient algorithms for division in the original semiring context.

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 $\mathbb{F}_2$.