Embeddable partial groups

This paper establishes a folklore theorem characterizing the embeddability of partial groups in groups via unique parenthesization of words, investigates examples of non-embeddable cases, and proves that a partial groupoid embeds in a groupoid if and only if its reduction embeds in a group.

The Gist

Here is an explanation of the paper "Embeddable Partial Groups" using simple language and creative analogies.

The Big Picture: The "Broken Calculator" Problem

Imagine you have a partial group. Think of this as a very picky, broken calculator.

• In a normal group (like a standard calculator), you can multiply any two numbers, and the order in which you group them doesn't matter. $(2 \times 3) \times 4$ is the same as $2 \times (3 \times 4)$.
• In a partial group, some buttons are broken. You can multiply some numbers, but not others. Sometimes, you can't even multiply three numbers in a row because the middle step is missing.

The Core Question:
Can we fix this broken calculator? Can we find a "perfect" calculator (a standard group) that contains our broken one, where all the rules still hold, but we just can't press the buttons that were originally broken?

If the answer is yes, we say the partial group is embeddable. If the answer is no, it's hopelessly broken.

The Main Discovery: The "One Path" Rule

The authors prove a famous "folklore" theorem (a rule everyone suspected but hadn't fully written down in this specific way).

The Rule: A partial group can be fixed (embedded in a perfect group) if and only if every possible sequence of numbers has at most one possible result, no matter how you group the operations.

The Analogy: The Hiking Trail
Imagine you are hiking from Point A to Point B.

• The Path: You have a map with trails. Some trails are paved (defined operations), and some are overgrown (undefined).
• The Problem: You want to know if there is a "Grand Canyon" (a perfect group) where this map fits perfectly.
• The Obstruction: Suppose you have a sequence of three trails: Trail 1, Trail 2, and Trail 3.
  • If you hike Trail 1 then Trail 2, you arrive at a campsite. Then you hike Trail 3.
  • If you hike Trail 2 then Trail 3, you arrive at a different campsite. Then you hike Trail 1.
  • If these two different hiking strategies lead to different final destinations, your map is broken. It contradicts the laws of geometry. You cannot fit this map into a perfect world.
• The Solution: The paper says: If every possible way of grouping your hikes leads to the same final destination (or if the path is impossible to complete), then your map is valid! It can be embedded in a perfect world.

The "Sad" and "Happy" Edges

The authors use some cute names to describe the parts of their mathematical maps:

• Happy Edges: These are the trails that behave well. If you take a "Happy Edge," it's clear where you are going, and it doesn't conflict with other paths.
• Sad Edges: These are the troublemakers. A "Sad Edge" is a trail that looks like it should be unique, but because of how the trails are grouped, it turns out to be the same as a completely different trail. This creates a contradiction.

The Verdict: If your map has no Sad Edges, it can be fixed. If it has even one, it's impossible to embed in a perfect group.

The "Universal Counterexamples" (The "Impossible Puzzles")

The authors didn't just say "it's broken"; they built the ultimate broken puzzles.

Imagine you want to prove that a specific type of puzzle is impossible to solve. You don't just show one bad puzzle; you build a "Master Puzzle" that represents every possible way a puzzle could fail.

• They constructed a specific mathematical object (called $N_{T,T'}$) for every possible way you can draw lines inside a polygon (triangulations).
• Think of this as a universal "glitch". If your partial group contains a copy of this "glitch," it is definitely broken.
• If your partial group does not contain any of these glitches, it is safe.

This allows mathematicians to look at a complex system and simply check: "Does it contain this specific glitch?" If yes, throw it away. If no, it's good to go.

The "Reduction" Trick: Flattening the Map

Finally, the paper tackles a harder version of the problem: Partial Groupoids.

• A Group is like a single city where everyone can travel to anyone.
• A Groupoid is like a country with many cities. You can travel from City A to City B, but maybe not from City A to City C.

The authors show a clever trick: Flatten the map.

• Imagine you take a map of a whole country (Groupoid) and glue all the cities together into one giant metropolis (Group). This is called "Reduction."
• The Big Insight: Your country-map is fixable if and only if the flattened metropolis-map is fixable.
• Why this matters: It turns a complicated, multi-city problem into a simple, single-city problem. If you can solve the puzzle for the metropolis, you've solved it for the whole country.

Summary

1. The Goal: Determine if a "broken" math system (partial group) can fit inside a "perfect" one.
2. The Test: Check if any sequence of moves has two different results depending on how you group them. If yes $\rightarrow$ Broken. If no $\rightarrow$ Fixable.
3. The Tool: They built "Universal Glitches" (counterexamples). If your system has a glitch, it's broken.
4. The Shortcut: To check a complex system with many objects, just glue them all together into one. If the "glued" version works, the original works too.

In short, the paper gives us a clear, step-by-step checklist to decide if a messy, incomplete mathematical structure can be saved and made perfect.

Technical Summary

Here is a detailed technical summary of the paper "Embeddable Partial Groups" by Philip Hackney, Justin Lynd, and Edoardo Salati.

1. Problem Statement

The paper addresses the fundamental question of embeddability for partial groups and partial groupoids.

• Context: A partial group (in the sense of Chermak) is a structure where multiplication is not defined for all pairs of elements, but only for specific "words" (sequences of elements).
• The Core Question: Under what conditions can a partial group be embedded as a subgroup (or sub-groupoid) of a genuine group (or groupoid)?
• The Obstruction: A known obstruction exists: if a word $w$ can be parenthesized in two different ways such that both parenthesizations yield valid products in the partial structure, but the results are distinct, the structure cannot be embedded in a group.
• The Gap: While this obstruction is necessary, it was a "folklore" conjecture whether it is also sufficient. Furthermore, the paper seeks to characterize the category of embeddable partial groupoids and understand the "degree" of non-embeddability using higher categorical invariants.

2. Methodology

The authors employ a combination of combinatorial group theory, simplicial homotopy theory, and category theory.

• Symmetric Sets and Simplicial Sets: The authors utilize the formalism of symmetric sets (functors $\Upsilon^{op} \to \mathbf{Set}$), which generalize simplicial sets to model invertibility. They treat partial groupoids as "spiny" symmetric sets (a specific type of edgy simplicial set).
• The Nerve and Fundamental Groupoid: They utilize the nerve functor $N: \mathbf{Cat} \to \mathbf{sSet}$ and its left adjoint $\tau: \mathbf{sSet} \to \mathbf{Cat}$ (the fundamental groupoid construction). For a symmetric set $X$, $\tau X$ represents the "free" groupoid generated by $X$.
• Word Reduction and Zig-Zags: The paper analyzes the equivalence relation on words generated by inner face maps (composition). They study "zig-zags" of words connecting two edges $f$ and $g$ to determine if they represent the same morphism in $\tau X$.
• Orthogonality Classes: The authors characterize embeddability using the concept of orthogonality in category theory. They construct specific "universal counterexamples" (partial groupoids) that detect non-embeddability.
• Rewrite Systems: In the final section, they use rewriting systems (reduction and serration of strings) to construct a monoid $M(C)$ associated with a category $C$, proving that the reduction of a partial groupoid embeds in this monoid.

3. Key Contributions and Results

A. The Embeddability Theorem (Folklore Confirmed)

The primary result is a rigorous proof of a folklore theorem (attributed to Baer and Tamari):

• Theorem: A partial groupoid $X$ embeds into a groupoid if and only if every word has at most one possible multiplication, regardless of the choice of parenthesization.
• Technical Formulation: This is equivalent to saying that the unit map $X \to N\tau X$ (from the partial groupoid to the nerve of its fundamental groupoid) is a monomorphism.
• Characterization: This occurs if and only if all 1-simplices (edges) in $X$ are "happy" (i.e., no two distinct edges are identified in the fundamental groupoid via a zig-zag of reductions).

B. Universal Counterexamples and Orthogonality

The authors construct a family of universal obstructions to embeddability:

• Construction: For any pair of triangulations $T, T'$ of a regular $(n+1)$-gon, they define a partial groupoid $N_{A_{T,T'}}$.
• Compatibility: They define "compatible" pairs of triangulations where the resulting structure is a valid partial groupoid.
• Orthogonality Characterization: A partial groupoid $X$ is embeddable if and only if it is orthogonal to the set of maps $N_{A_{T,T'}} \to A_{T,T'}$ for all "well-behaved" pairs of triangulations. This concretely describes the category of embeddable partial groupoids as an orthogonality class.

C. Degree of Non-Embeddability

The paper computes the degree of these universal counterexamples, an invariant related to higher Segal spaces:

• Result: The degree of $N_{A_{T,T'}}$ is 2 if the pair of triangulations $(T, T')$ does not admit a "cone" (a specific geometric configuration of triangles). If a cone exists, the degree is 3.
• Significance: This links the combinatorial structure of the triangulations (Tamari lattice) to the homotopical complexity of the partial groupoid.

D. Reduction and Embeddability

The paper establishes a crucial relationship between partial groupoids and partial groups (the "reduced" case where all vertices are identified):

• Theorem: A partial groupoid $X$ is embeddable if and only if its reduction $RX$ (a partial group) is embeddable.
• Proof Mechanism: They construct a monoid $M(C)$ from a category $C$ using a confluent rewrite system (reducing identities and composing composable non-identities). They show that the reduction of a partial groupoid embeds into the group $M(\tau X)$.

4. Significance

• Resolution of Open Problems: The paper settles the long-standing question of whether the "unique multiplication" condition is sufficient for embeddability, confirming the folklore theorem for both partial groups and partial groupoids.
• Categorical Framework: It provides a robust categorical framework (using symmetric sets and orthogonality) for studying partial algebraic structures, moving beyond ad-hoc combinatorial arguments.
• Connection to Fusion Systems: The results have direct implications for $p$-local group theory. Linking systems (which model saturated fusion systems) are special partial groups. The paper's results help clarify the relationship between the fundamental groups of these linking systems and their underlying partial structures (e.g., confirming that elements in certain linking localities are associated by words).
• Generalization: The work generalizes classical results by Baer and Tamari (originally for one-object structures) to the many-object setting (groupoids), utilizing modern tools like the Gabriel-Zisman reflection and associahedra.

In summary, this paper provides a complete characterization of when partial groups embed into groups, utilizing advanced simplicial techniques to construct universal obstructions and proving that the embeddability of a structure is entirely determined by its "reduced" (single-object) counterpart.