Bibliographic record
Abstract
Mathematicians attach importance to extending ideals in algebraic structures. The concept of bi-quasi (ƁԚ) ideal was introduced as a generalization version of quasi-ideal, bi-ideal, and left (right) ideals in semigroups. This paper applies this concept to soft set theory and semigroups, introducing the notion of "Soft union (S-uni) ƁԚ ideal." The aim of this paper is to explore the relationships between S-uni ƁԚ ideals and other types of S-uni ideals in semigroups. It is shown that every S-uni bi-ideal, S-uni ideal, S-uni quasi-ideal, and S-uni interior ideal of an idempotent soft set are S-uni ƁԚ ideals. Counterexamples demonstrate that the converses are not always true unless the semigroup is special soft simple or regular. For special soft simple semigroups, the S-uni ƁԚ ideal coincides with the S-uni bi-ideal, S-uni left (right) ideal, and S-uni quasi-ideal. Additionally, we provide conceptual definitions and analyses of the new concept in the context of soft set operations, supporting our claims with clear examples.
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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 | 0.002 | 0.002 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.001 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.001 | 0.004 |
| Scholarly communication | 0.002 | 0.004 |
| Open science | 0.000 | 0.002 |
| Research integrity | 0.000 | 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".