Generalized Elementary Tietze Transformations (GETTs)

• GETTs are generalized transformation operations that extend classical combinatorial group theory to arbitrary monoids by using rewriting systems.
• They enable conversion between convergent presentations through moves like redundant rule addition/removal, generator introduction, and collapse of confluent subsystems.
• GETTs facilitate algorithmic analysis, streamline the resolution of word problems, and broaden the applicability of rewriting methods in algebra and categorical frameworks.

A Generalized Elementary Tietze Transformation (GETT) is a foundational concept in monoidal rewriting theory for arbitrary monoids, extending the classical Tietze transformations of combinatorial group theory. GETTs provide the basis for a global classification of presentations of monoids via rewriting systems that need not originate from free objects. This notion is pivotal for understanding the equivalence classes of finite, complete presentations of a monoid under operations that preserve the presented algebraic structure (Magalhães, 15 Jan 2026). Below, the essential technical framework and the algebraic and categorical implications of GETTs are detailed within the context of monoidal rewriting systems.

1. Monoidal Rewriting Systems on Arbitrary Monoids

A monoidal rewriting system (MRS) on a fixed monoid $(M, \cdot)$ consists of a binary relation $R \subseteq M \times M$ interpreted as oriented reduction steps. The one-step reduction $u \to_R v$ is defined as: $u = x s y \to_R v = x t y$ iff $(s, t) \in R$ and $x, y \in M$. This abstract structure subsumes classical string rewriting systems, which arise when $M$ is a free monoid and $R$ encodes substitutions. Unlike free monoids, arbitrary monoids may possess nontrivial algebraic structure—such as closure operators, topos-theoretic subobject lattices, or module operations—which are not first-order axiomatizable in the same way (Magalhães, 15 Jan 2026).

Central properties in this setting include:

• Termination (Noetherianity): No infinite chains $m_0 \to_R m_1 \to_R m_2 \to_R \cdots$ exist.
• Confluence: For every $m \to_R^* m_1$, $m \to_R^* m_2$, there exists $m'$ such that $m_1 \to_R^* m'$ and $m_2 \to_R^* m'$.
• Normal Form Existence: In a Noetherian confluent MRS, each $m \in M$ possesses a unique irreducible representative $\overline{m}$, and the set of all irreducibles $\overline{M}$ forms a monoid under the induced product $\overline{u}\star\overline{v} = \overline{u v}$.

2. The 2-Category of Noetherian Confluent MRS

This conceptual expansion is formalized by the strict 2-category $\mathbf{NCRS}_2$:

• Objects: Noetherian, confluent MRS $(M, R)$.
• 1-Morphisms: Monoid homomorphisms $\phi: (M, R) \to (N, S)$ with the property that $(u, v) \in R \implies \phi(u) \to_S^* \phi(v)$.
• 2-Morphisms: Uniquely determined when for every $m \in M$, $\phi(m) \leftrightarrow_S^* \psi(m)$; forming a thin groupoid structure.

This category admits a canonical biadjunction with the category of monoids:

• The 'presentation' 2-functor $G$ assigns to any monoid $M$ a canonical convergent MRS $G(M) = (F_{M^+}, R_M)$, where $F_{M^+}$ is the free monoid on $M^+$ and $R_M$ encodes the multiplication of $M$.
• The 'irreducible' 2-functor $I$ maps an MRS $(M, R)$ to the monoid $(\overline{M}, \star)$ of irreducibles.
• The biadjunction $G \dashv I$ is realized via functorial extensions and identifications of normal forms under these constructions.

3. Definition and Types of GETTs

Generalized Elementary Tietze Transformations provide the transformations which connect any two convergent presentations of a fixed monoid $M$:

1. Addition of Redundant Rules: If $a \leftrightarrow_R^* b$, augment $R$ by $(a, b)$. This does not alter the irreducibles.
2. Removal of Redundant Rules: If $(a, b) \in R$, but $a \leftrightarrow_{R \setminus \{(a, b)\}}^* b$, remove $(a, b)$.
3. Introduction of Generators: Add a new generator $v$ freely and impose $(v, a)$, effectively extending $M$ without changing its quotient structure.
4. Collapse of Confluent Subsystem: Given a subset $J \subseteq R$ that is confluent, replace $(A, R)$ by the induced MRS on the irreducibles of $(A, J)$, with rules inherited from $R \setminus J$, provided the resultant system remains convergent.

These moves, individually or in sequence, transform a presentation without changing its monoid of irreducibles ($I(A, R) \simeq M$), and hence preserve the algebraic object presented.

4. Classification Theorem via GETTs

The central structural result is the classification theorem:

Let $(A, R)$ and $(B, S)$ be convergent MRSs presenting the same monoid $M$. There exists a (possibly infinite) sequence of GETTs connecting $(A, R)$ to $(B, S)$.

The sequence is constructed by:

• Collapsing $(A, R)$ to the trivial presentation $(M, \emptyset)$ using transformation (4).
• Modifying generator sets via (3) and rule sets via (1), (2) to recover the canonical presentation $G(M)$.
• Finally, collapsing $G(M)$ to $(B, S)$ as in the first step.

This establishes that GETT-equivalence is a complete invariant of convergent presentations of $M$ in the category $\mathbf{NCRS}_2$. Hence, the moduli space of presentations is determined up to GETTs.

5. Logical and Categorical Implications

Working with GETTs and MRS over arbitrary monoids circumvents the limitations of first-order logic, accommodating algebraic theories not reducible to presentations over free monoids. This is particularly relevant for:

• Logical frameworks: Internalization of rewriting for model-theoretic purposes.
• Topos-theoretic scenarios: Closure operators and subobject lattices as MRSs.
• High-dimensional rewriting: Polygraphic presentations and higher categories constructed over arbitrary algebraic bases.

The GETT framework facilitates the extension of methods such as Knuth–Bendix completion, critical-pair analysis, and normal-form algorithms to arbitrary algebraic settings (Magalhães, 15 Jan 2026).

6. Practical Consequences and Research Directions

The adoption of GETTs in the context of MRS supports:

• Algorithmic comparisons between distinct presentations for efficiency or computational suitability.
• Existence of normal forms and canonical representatives for elements of $M$, streamlining the word problem in non-free settings.
• Rigorous classification and transformation of algebraic presentations across categorical and logical domains.

GETTs provide the necessary moves for a global theory of presentations, enhancing the transfer, simplification, and study of rewriting systems on monoids with nontrivial structure. Their categorical correspondence ensures that the collection of Noetherian, confluent presentations is well-understood up to GETT sequences, generalizing the combinatorial group theory paradigm to a broader algebraic spectrum.