Generalized Soft Union Bi-Ideals and Interior Ideals of Semigroups: Soft Union Bi-Interior Ideals of Semigroups
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Bibliographic record
Abstract
Generalizing the ideals of an algebraic structure has shown to be both beneficial and interesting for mathematicians. In this context, the idea of the bi-interior ideal was introduced as a generalization of the bi-ideal and interior ideal of a semigroup. By introducing "soft union (S-uni) bi-interior ideals of semigroups", we apply this idea to semigroups and soft set theory in this study. Finding the relationships between S-uni bi-interior ideals and other specific kinds of S-uni ideals of a semigroup is the main aim of this study. Our results show that an S-uni bi-interior ideal is an S-uni subsemigroup of a special soft simple semigroup, and that the S-uni bi-interior ideal of semigroup is a generalization of the S-uni left (right/two-sided) ideal, bi-ideal, interior ideal, and quasi-ideal, however, the converses are not true with counterexamples. We demonstrate that the semigroup should be a special soft simple semigroup in order to satisfy the converses. Furthermore, we present conceptual characterizations and analysis of the new concept in terms of regarding soft set operations and notions supporting our assertions with particular, illuminating examples.
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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 | 0.004 | 0.001 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.002 | 0.002 |
| 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