Rewriting Systems on Arbitrary Monoids

In this paper, we introduce monoidal rewriting systems (MRS), an abstraction of string rewriting in which reductions are defined over an arbitrary ambient monoid rather than a free monoid of words. This shift is partly motivated by logic: the class of free monoids is not first-order axiomatizable, so “working in the free setting” cannot be treated internally when applying first-order methods to rewriting presentations. To analyze these systems categorically, we define $\mathbf{NCRS_2}$ as the 2-category of Noetherian Confluent MRS. We then prove the existence of a canonical biadjunction between $\mathbf{NCRS_2}$ and $\mathbf{Mon}$. Finally, we classify all Noetherian Confluent MRS that present a given fixed monoid. For this, we introduce Generalized Elementary Tietze Transformations (GETTs) and prove that any two presentations of a monoid are connected by a (possibly infinite) sequence of these transformations, yielding a complete characterization of generating systems up to GETT-equivalence.

💡 Research Summary

The paper introduces Monoidal Rewriting Systems (MRS) as a generalisation of classical string rewriting systems (SRS) that operate not on a free monoid of words but directly on an arbitrary ambient monoid. The motivation is logical: free monoids are not first‑order axiomatizable, so when one wishes to study rewriting presentations using first‑order or model‑theoretic tools, the “freeness” condition cannot be internalised. By defining rewriting rules as a binary relation R ⊆ M × M on any monoid (M,·), the authors decouple rewriting from the syntactic setting of words and obtain a notion of reduction that is intrinsic to the algebraic structure.

The paper proceeds in several stages:

1. Foundations and Basic Lemmas – After fixing notation, the authors define the one‑step reduction a →_R b (allowing arbitrary left and right contexts) and prove elementary lemmas that mirror the classical SRS theory: compatibility of →_R with monoid multiplication, that the symmetric‑reflexive‑transitive closure ↔_R is a congruence, and that in a Noetherian‑confluent MRS each element has a unique normal form (denoted \bar a). They show that the set of normal forms equipped with a new product \bar· (obtained by multiplying representatives and then normalising) forms a monoid, which is isomorphic to the quotient M/↔_R. These results demonstrate that the core rewriting properties (Church–Rosser, uniqueness of normal forms) survive without any reliance on freeness.

2. Examples – Two illustrative examples are given: (i) the natural numbers with addition and a single rule (2,0) yielding a Z/2Z quotient, and (ii) the powerset monoid of a three‑element set with rules that add elements under specific conditions. Both are Noetherian and confluent, showing that MRS can capture non‑trivial algebraic behaviours.

3. 2‑Categorical Framework – The authors construct a strict 2‑category NCRS₂ whose objects are Noetherian‑confluent MRS, 1‑cells are strong transformations (essentially monoid homomorphisms preserving the rewriting relation), and 2‑cells are modifications between such transformations. They also consider the strict 2‑category Mon of monoids, monoid homomorphisms, and natural transformations. The main categorical result is a canonical biadjunction L ⊣ R between NCRS₂ and Mon:

   • The left 2‑functor L sends an MRS (M,R) to its quotient monoid M/↔*_R.
   • The right 2‑functor R sends a monoid N to the “free” MRS (N,∅) (i.e., the same underlying set with no rewriting rules).
   • The unit η and counit ε are strong natural transformations, and the triangle identities hold up to invertible modifications (triangulator modifications) that satisfy the swallow‑tail identities. This biadjunction formalises the intuition that MRS are presentations of monoids and that passing between presentations and underlying monoids is a universal construction in the 2‑categorical sense.

4. Generalised Elementary Tietze Transformations (GETT) – Classical Tietze transformations provide a finite set of moves that relate two finite SRS presentations of the same monoid. The authors generalise this to GETT, a set of four elementary moves applicable to any MRS:

   1. Adding a rule that is already a consequence of the current congruence.
   2. Deleting a rule whose removal does not change the induced congruence.
   3. Introducing a new generator together with a rule linking it to an existing element.
   4. Removing a generator together with its defining rule, provided the remaining system still presents the same monoid. Each move preserves the presented monoid, and the authors prove a GETT‑equivalence theorem: any two Noetherian‑confluent MRS presenting the same monoid can be connected by a (possibly infinite) sequence of GETT moves. The proof proceeds by normalising both systems to a canonical form (where all rules involve irreducible elements) and then systematically aligning their generators and rules using the four moves. The allowance of infinite sequences is essential because, unlike the classical finite case, some monoids admit presentations that necessarily involve infinitely many generators or rules.

5. Implications and Future Work – By removing the dependence on free monoids, MRS open the door to applying first‑order logical methods directly to rewriting presentations. The biadjunction gives a clean categorical semantics for presentations, while GETT provides a complete algebraic calculus for transforming presentations. The authors suggest further research directions such as studying convergence properties of infinite GETT chains, extending the framework to groups, semigroups, or higher‑dimensional algebraic structures, and investigating algorithmic aspects of deciding GETT‑equivalence.

Overall, the paper makes three substantial contributions: (i) a robust definition of rewriting over arbitrary monoids preserving the essential confluence and termination theory, (ii) a 2‑categorical biadjunction linking presentations to monoids, and (iii) a generalized Tietze‑type calculus that fully characterises presentation equivalence. These results bridge rewriting theory, categorical algebra, and logical foundations, and they promise new tools for both theoretical investigations and practical applications such as automated theorem proving and algebraic specification.