# Topological rewriting systems applied to standard bases and syntactic algebras

## TL;DR

This paper introduces a topological approach to rewriting systems on vector spaces, linking confluence properties to lattice operations and standard bases, and explores duality in syntactic algebras.

## Contribution

It presents a novel topological framework for rewriting systems, characterizes confluence via lattice operations, and connects these concepts to standard bases and syntactic algebras.

## Key findings

- Topological confluence is characterized by lattice operations.
- Standard bases induce topologically confluent rewriting systems.
- Duality helps determine if an algebra is syntactic.

## Abstract

We propose a functional description of rewriting systems on topological vector spaces. We introduce the topological confluence property as an approximation of the confluence property. Using a representation of linear topological rewriting systems with continuous reduction operators, we show that the topological confluence is characterised by lattice operations. We relate these operations to standard bases and show that the latter induce topologically confluent rewriting systems on formal power series. Finally, we investigate duality for reduction operators that we relate to series representations and syntactic algebras. In particular, we use duality for proving that an algebra is syntactic or not.