Dynamics of implicit operations and tameness of pseudovarieties of groups
Bibliographic record
Abstract
This work gives a new approach to the construction of implicit operations. By considering “higher-dimensional” spaces of implicit operations and implicit operators between them, the projection of idempotents back to one-dimensional spaces produces implicit operations with interesting properties. Besides providing a wealth of examples of implicit operations which can be obtained by these means, it is shown how they can be used to deduce from results of Ribes and Zalesskiĭ, Margolis, Sapir and Weil, and Steinberg that the pseudovariety of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p"> <mml:semantics> <mml:mi>p</mml:mi> <mml:annotation encoding="application/x-tex">p</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -groups is tame. More generally, for a recursively enumerable extension closed pseudovariety of groups <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="bold upper V"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="bold">V</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathbf {V}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , if it can be decided whether a finitely generated subgroup of the free group with the pro- <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="bold upper V"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="bold">V</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathbf {V}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> topology is dense, then <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="bold upper V"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="bold">V</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathbf {V}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is tame.
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How this classification was reachedexpand
Full frame distilled prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. Learned from the 10,348 direct Codex labels and 10,348 direct Gemma labels. Candidate is the union of thresholded teacher heads; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels or direct frontier model labels.
Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.000 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.000 |
| Bibliometrics | 0.000 | 0.001 |
| Science and technology studies | 0.000 | 0.002 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.000 | 0.000 |
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 teacher head, 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".