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Embeddable partial groups

Philip Hackney, Justin Lynd, Edoardo Salati

TL;DR

This work formalizes when a partial groupoid (and hence a partial group) can embed into a group, showing that embeddability is equivalent to the absence of words with multiple, choice-dependent multiplications, and extends the criterion to many-object settings via the reflection $\tau$. It introduces universal counterexamples built from triangulations (NA^{T,T'}, A^{T,T'}) and proves an orthogonality characterization: an object is embeddable iff it is orthogonal to the maps $NA^{T,T'}\to A^{T,T'}$ for well-behaved triangulation pairs. The paper then connects this to a degree invariant of symmetric sets, showing deg$(NA^{T,T'})=3$ precisely when a cone exists between triangulations, tying embeddability to higher-Segal properties, and finally proves that embeddability is determined by reduction via a rewrite-system approach and a monoid construction $M(C)$; these results interface with broader structures in $p$-local group theory and linking systems.

Abstract

We record a folklore theorem that says a partial group embeds in a group if and only if each word has at most one possible multiplication, regardless of choice of parenthesization. We further investigate the partial groups which are exemplars of non-embeddability. Finally we show that a partial groupoid embeds in a groupoid if and only if its reduction embeds in a group.

Embeddable partial groups

TL;DR

This work formalizes when a partial groupoid (and hence a partial group) can embed into a group, showing that embeddability is equivalent to the absence of words with multiple, choice-dependent multiplications, and extends the criterion to many-object settings via the reflection . It introduces universal counterexamples built from triangulations (NA^{T,T'}, A^{T,T'}) and proves an orthogonality characterization: an object is embeddable iff it is orthogonal to the maps for well-behaved triangulation pairs. The paper then connects this to a degree invariant of symmetric sets, showing deg precisely when a cone exists between triangulations, tying embeddability to higher-Segal properties, and finally proves that embeddability is determined by reduction via a rewrite-system approach and a monoid construction ; these results interface with broader structures in -local group theory and linking systems.

Abstract

We record a folklore theorem that says a partial group embeds in a group if and only if each word has at most one possible multiplication, regardless of choice of parenthesization. We further investigate the partial groups which are exemplars of non-embeddability. Finally we show that a partial groupoid embeds in a groupoid if and only if its reduction embeds in a group.
Paper Structure (4 sections, 15 theorems, 25 equations, 3 figures)

This paper contains 4 sections, 15 theorems, 25 equations, 3 figures.

Table of Contents

  1. Embeddability
  2. Orthogonality set
  3. Degree of NATT
  4. Reduction and embeddability

Key Result

Theorem 2

Let $X$ be a partial groupoid and $f,g\in X_1$. If $[f] = [g]$ in $\tau X$, then there is a word $w \in \mathbf{W}^+(X)$ with $f \mathrel{\reflectbox{$⇝$}} w \rightsquigarrow g$.

Figures (3)

  • Figure 1: Flip graphs showing different triangulations. Image by TheMathCat at English Wikipedia, licensed under CC BY 4.0. https://en.wikipedia.org/wiki/File:Flip_graphs.svg
  • Figure 2: Incompatible triangulations of the pentagon
  • Figure 3: Compatible triangulations of the pentagon and $\operatorname{NA}^{T,T'}$

Theorems & Definitions (38)

  • Example 1
  • Theorem 2
  • proof
  • Corollary 3
  • proof
  • Remark 4
  • Definition 5
  • Theorem 6
  • proof
  • Example 7
  • ...and 28 more