The action of an inverse semigroup on its Stone-Čech compactification
Bibliographic record
Abstract
We initiate the study of the Stone-Čech transformation groupoid $\mathcal{G} = \mathcal{S}\ltimesβ\mathcal{S}$ of an inverse semigroup $\mathcal{S}$. We prove that the properties of being Hausdorff, principal, and effective are all equivalent for $\mathcal{G}$, and give an algebraic condition on $\mathcal{S}$ equivalent to the Hausdorffness of $\mathcal{G}$. We show that the Hausdorffness of Exel's tight groupoid $\mathcal{G}_{\text{tight}}(\mathcal{S})$ is necessary for the Hausdorffness of $\mathcal{G}$. Finally, we clarify the connection between several crossed product constructions involving this groupoid, and show that when it is Hausdorff and $\mathcal{S}$ has the Property (FL) of Lledó and Martínez, then $\mathcal{G}$ is amenable if and only if the reduced C$^*$-algebra $\text{C}^*_r(\mathcal{S})$ is exact.
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How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.000 | 0.001 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.001 |
| Bibliometrics | 0.001 | 0.000 |
| Science and technology studies | 0.001 | 0.002 |
| Scholarly communication | 0.002 | 0.002 |
| Open science | 0.000 | 0.001 |
| Research integrity | 0.000 | 0.001 |
| Insufficient payload (model declined to judge) | 0.004 | 0.001 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from itClassification
machine, unvalidatedMachine predicted; a candidate call from one source (direct Gemma or distilled Codex), not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".