Bibliographic record
Abstract
Abstract A magma ( M , *) is a nonempty set with a binary operation. A double magma ( M , *, •) is a nonempty set with two binary operations satisfying the interchange law ( w * x ) • ( y * z ) = ( w • y )*( x•z ). We call a double magma proper if the two operations are distinct, and commutative if the operations are commutative. A double semigroup , first introduced by Kock, is a double magma for which both operations are associative. Given a non-trivial group G we define a system of two magma ( G , *, •) using the commutator operations x * y = [ x, y ](= x −1 y −1 x y ) and x • y = [ y, x ]. We show that ( G , *, •) is a double magma if and only if G satisfies the commutator laws [ x, y; x, z ] = 1 and [ w, x; y, z ] 2 = 1. We note that the first lawdefines the class of 3-metabelian groups. If both these laws hold in G , the double magma is proper if and only if there exist x 0 , y 0 ∊ G for which [x 0 , y 0 ] 2 ≠ 1. This double magma is a double semigroup if and only if G is nilpotent of class two. We construct a specific example of a proper double semigroup based on the dihedral group of order 16. In addition, we comment on a similar construction for rings using Lie commutators.
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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 | 0.002 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.003 | 0.002 |
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; both teacher heads agree on what is shown here.
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".