A Study of Non-Associative Ordered Semigroups in Terms of Semilattices via Smallest (Double-Framed Soft) Ideals
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Bibliographic record
Abstract
Soft set theory, introduced by Molodtsov has been considered as a successful mathematical tool for modeling uncertainties. A double-framed soft set is a generalization of a soft set, consisting of union soft sets and intersectional soft sets. An ordered AG-groupoid can be referred to as a non-associative ordered semigroup, as the main difference between an ordered semigroup and an ordered AG-groupoid is the switching of an associative law. In this paper, we define the smallest left (right) ideals in an ordered AG-groupoid and use them to characterize a strongly regular class of a unitary ordered AG-groupoid along with its semilattices and double-framed soft (briefly DFS) l-ideals (r-ideals). We also give the concept of an ordered A* G**-groupoid and investigate its structural properties by using the generated ideals and DFS l-ideals (r-ideals). These concepts will verify the existing characterizations and will help in achieving more generalized results in future works.
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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 | 0.002 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.000 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.001 | 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