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−1x 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 x0, y0 ∊ G for which [x0 , y0]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 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 | 0.001 | 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 | 0.001 | 0.002 |
| Open science | 0.000 | 0.002 |
| Research integrity | 0.001 | 0.001 |
| Insufficient payload (model declined to judge) | 0.003 | 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".