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Rewriting Systems on Arbitrary Monoids

This paper introduces monoidal rewriting systems (MRS) as an abstraction of string rewriting over arbitrary ambient monoids to address logical limitations of free monoids, and establishes a canonical biadjunction between the 2-category of Noetherian Confluent MRS and the category of monoids while classifying all such systems presenting a fixed monoid via Generalized Elementary Tietze Transformations.

Original authors: Eduardo Magalhães

Published 2026-02-02

📖 6 min read🧠 Deep dive

CC BY 4.0

Original authors: Eduardo Magalhães

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are trying to solve a puzzle where you have a set of rules for changing one thing into another. In the world of computer science and math, this is usually done with strings of letters (like words in a dictionary). If you have the word "cat" and a rule that says "cat" becomes "dog," you can swap them. This is the traditional way of doing things, called String Rewriting.

However, the author of this paper, Eduardo Magalhães, asks a simple but profound question: What if we aren't just playing with words? What if we are playing with numbers, shapes, or even abstract ideas that don't look like words at all?

Here is a breakdown of the paper's main ideas using everyday analogies:

1. The Problem: Being Too Picky About "Words"

Traditionally, rewriting systems only work on Free Monoids. Think of a Free Monoid as a giant, empty warehouse where you can only stack boxes (letters) in a line. You can only combine them by sticking them together.

The Issue: The paper argues that this is too limiting. It's like saying you can only rearrange furniture if you are in a warehouse with no walls. In the real world (and in logic), we often deal with structures that have their own internal rules (like a clock where 12 + 1 = 1, or a group of friends where "Alice + Bob" is just "The Group").
The Logic Gap: The author points out that "being a free warehouse" is a very specific, hard-to-define rule in the language of logic. If you want to study these systems using standard logical tools, you get stuck because you can't easily define "free" inside the system itself.

2. The Solution: Monoidal Rewriting Systems (MRS)

The author introduces Monoidal Rewriting Systems (MRS).

The Analogy: Instead of just rearranging letters in a line, imagine you have a toolbox (a Monoid). This toolbox has a specific way of combining tools (multiplication).
- In a string system, you can only glue "A" and "B" to make "AB".
- In an MRS, you can combine any two items in your toolbox, provided they follow the toolbox's rules. Maybe your toolbox is a set of numbers where you add them, or a set of shapes where you overlap them.
The Shift: The paper says, "Let's stop pretending everything is a word. Let's let the rules work directly on the objects themselves." This makes the system more flexible and "internal" to the structure it describes.

3. The "Perfect" State: Noetherian and Confluent

In any rewriting game, you want two things to happen:

Noetherian (Termination): The game must eventually end. You can't keep changing things forever in a loop. (e.g., You can't have a rule that turns "A" into "B" and "B" back into "A" forever).
Confluent (Consistency): No matter which order you apply the rules, you should end up with the same final result. (e.g., If you have a messy room, it shouldn't matter if you pick up the socks first or the books first; the room should end up clean in the same way).

When a system has both of these, you can take any messy input and reduce it to a unique "Normal Form" (the cleanest, simplest version of that object).

4. The Big Connection: The "Translator" (Biadjunction)

The paper builds a bridge between two worlds:

World A: The messy, rule-heavy world of Rewriting Systems (MRS).
World B: The clean, simple world of Monoids (the final structures).

The author creates a Translator (a mathematical tool called a biadjunction) that works both ways:

From Rules to Structure: If you have a set of rules, the translator finds the "clean" structure hidden inside them (the Monoid of irreducibles).
From Structure to Rules: If you have a clean structure (like the number 5), the translator can build a "canonical" set of rules that generates it.

The Metaphor: Imagine you have a sculpture (the Monoid).

One way to describe it is to say, "It's made of clay." (The Structure).
Another way is to give a list of instructions: "Take a lump, flatten it, cut a circle, smooth the edges." (The Rewriting System).
The paper proves that these two descriptions are perfectly linked. You can go from the instructions to the sculpture, and from the sculpture back to the best possible set of instructions, without losing any information.

5. The "Tietze" Transformations: The Magic Wands

Finally, the paper answers a tricky question: "If I have two different sets of rules that both build the same sculpture, how are they related?"

In the old world of string rewriting, there was a famous set of moves called Tietze Transformations that could turn one set of rules into another. The author invents Generalized Elementary Tietze Transformations (GETTs) for this new, broader world.

The Analogy: Imagine you have two different recipes for making a cake.
- Recipe A says: "Mix flour, sugar, eggs."
- Recipe B says: "Mix dry ingredients, then wet ingredients, then bake."
- Even though the steps look different, they make the same cake.
The Result: The paper proves that you can transform any valid recipe (Noetherian Confluent MRS) into any other valid recipe for the same cake by using a sequence of these "GETT moves."
- Move 1: Add a rule that is already true (redundant).
- Move 2: Remove a rule that is already covered by others.
- Move 3: Introduce a new ingredient (symbol) to help explain a step.
- Move 4: A complex move that simplifies the whole system by focusing on a specific part of the rules.

Summary

This paper takes the concept of "rewriting" (changing things based on rules) and frees it from the constraint of "words." It shows that:

You can do this on any mathematical structure, not just strings.
There is a perfect, logical bridge between the rules and the result.
Any two sets of rules that produce the same result can be transformed into each other using a specific, universal set of moves.

It's a bit like realizing that while you can describe a house by listing its bricks (strings), you can also describe it by its architectural blueprint (monoid), and you can mathematically prove that every blueprint has a unique, perfect set of instructions to build it, and every set of instructions leads to a unique blueprint.

Technical Summary: Rewriting Systems on Arbitrary Monoids

Problem Statement
The study of rewriting systems has historically been dominated by String Rewriting Systems (SRS), where reductions are defined over free monoids generated by an alphabet. While SRS provides a robust framework for solving the word problem and analyzing algebraic structures, it possesses a structural limitation: it is intrinsically tied to free monoids. From a logical perspective, this creates a foundational mismatch. The class of free monoids is not first-order axiomatizable; consequently, when applying first-order or model-theoretic methods to rewriting presentations, the property of "freeness" cannot be treated internally but must be imposed as an external semantic condition. Furthermore, classical rewriting relies heavily on the syntactic representation of words, which may not be strictly necessary for the underlying algebraic properties of reduction, such as confluence and termination.

Methodology
To address these limitations, the paper introduces Monoidal Rewriting Systems (MRS), an abstraction where reductions are defined directly over an arbitrary ambient monoid $(M, \cdot)$ rather than a free monoid of words. An MRS is defined as a triple $(M, \cdot, R)$ , where $R \subseteq M \times M$ is a binary relation. The reduction relation $a \to_R b$ is defined by the existence of $x, y, s, t \in M$ such that $a = xsy$, $b = xty$, and $(s, t) \in R$ .

The methodology proceeds through three main stages:

Algebraic Generalization: The paper establishes that fundamental properties of SRS (such as the Church-Rosser property, Noetherianity, and the existence of unique normal forms) generalize directly to MRS. It proves that for a Noetherian confluent MRS, the set of irreducible elements forms a monoid isomorphic to the quotient monoid $M / \leftrightarrow^*_R$ .
Categorical Formulation: The systems are organized within a strict 2-categorical framework. The author defines $\mathbf{NCRS}_2$ as the 2-category of Noetherian Confluent MRS, where 0-cells are the systems, 1-cells are homomorphisms preserving the reduction structure, and 2-cells represent equivalences between homomorphisms. This structure is compared to the 2-category of monoids, $\mathbf{Mon}$ .
Transformation Theory: To classify the family of MRS presenting a fixed monoid, the paper introduces Generalized Elementary Tietze Transformations (GETTs). These are four types of operations (adding/removing rules, adding new generators, and collapsing coherent subsets of rules) adapted to the arbitrary monoid setting.

Key Contributions and Results

Definition of MRS and Fundamental Properties: The paper formalizes MRS and demonstrates that the syntactic constraints of free monoids are not required for the core lemmas of rewriting theory. It proves that the quotient of an MRS by its congruence relation is isomorphic to the monoid of its irreducible elements.
Biadjunction between $\mathbf{NCRS}_2$ and $\mathbf{Mon}$ : A central result is the existence of a canonical biadjunction between the 2-category of Noetherian Confluent MRS ( $\mathbf{NCRS}_2$ $NCRS_{2}$ ) and the 2-category of monoids ( $\mathbf{Mon}$ $Mon$ ).
- The right adjoint $I: \mathbf{NCRS}_2 \to \mathbf{Mon}$ maps a system to its monoid of irreducibles.
- The left adjoint $G: \mathbf{Mon} \to \mathbf{NCRS}_2$ constructs a "canonical" Noetherian confluent MRS for any given monoid $M$ by utilizing the free monoid generated by $M \setminus \{1\}$ and encoding the monoid multiplication as reduction rules.
- This biadjunction establishes a universal property: maps from a canonical presentation $G(M)$ to any system $A$ correspond uniquely (up to 2-cell isomorphism) to monoid homomorphisms from $M$ to the monoid of irreducibles of $A$ .
Generalized Tietze Theorem: The paper generalizes Tietze's Theorem to the setting of arbitrary monoids. It defines four types of GETTs and proves that any two Noetherian confluent MRS presenting the same monoid are connected by a (possibly infinite) sequence of these transformations. Specifically, if two systems present the same monoid $M$ , one can be transformed into the other using at most $|B|$ GETTs, where $|B|$ is the cardinality of the underlying set of the target system.

Significance
The paper claims that this work provides a complete characterization of generating systems for a monoid up to GETT-equivalence, removing the reliance on free monoids. By shifting the setting from free monoids to arbitrary monoids, the framework allows rewriting systems to be treated internally within first-order logic and model theory, as the "freeness" constraint is no longer a prerequisite for the definition of the system.

Furthermore, the 2-categorical organization of these systems offers a higher-dimensional perspective on rewriting. It treats generating systems not merely as static presentations but as dynamic structures where equivalences between systems can be studied using categorical tools. The biadjunction result specifically clarifies the relationship between "semantic" monoid maps and "syntactic" maps of rewriting systems, showing that the latter are uniquely determined by the former up to the inherent 2-dimensional structure of $\mathbf{NCRS}_2$ .

The work also provides diverse examples illustrating the versatility of MRS, including applications to topological spaces (closure operators), elementary toposes (Lawvere-Tierney topologies), and Horn theories in logic, demonstrating that the monoidal structure is the essential component for rewriting, rather than the specific syntax of words.