Multiplicatively collapsing and rewritable algebras
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Bibliographic record
Abstract
A semigroup $S$ is called $n$-collapsing if, for every $a_1,\ldots , a_n$ in $S$, there exist functions $f\neq g$ (depending on $a_1,\ldots , a_n$), such that \[ a_{f(1)}\cdots a_{f(n)} = a_{g(1)}\cdots a_{g(n)};\] it is called collapsing if it is $n$-collapsing, for some $n$. More specifically, $S$ is called $n$-rewritable if $f$ and $g$ can be taken to be permutations; $S$ is called rewritable if it is $n$-rewritable for some $n$. Semple and Shalev extended Zelmanovâs solution of the restricted Burnside problem by proving that every finitely generated residually finite collapsing group is virtually nilpotent. In this paper, we consider when the multiplicative semigroup of an associative algebra is collapsing; in particular, we prove the following conditions are equivalent, for all unital algebras $A$ over an infinite field: the multiplicative semigroup of $A$ is collapsing, $A$ satisfies a multiplicative semigroup identity, and $A$ satisfies an Engel identity. We deduce that, if the multiplicative semigroup of $A$ is rewritable, then $A$ must be commutative.
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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 | 0.001 | 0.002 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.001 |
| 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 it