Bibliographic record
Abstract
The concept of tri-quasi ideal was presented as a generalization of quasi-ideal, interior ideal, and bi-ideal. In this paper, we transfer this concept to soft set theory and semigroups,and introduce a novel type of soft union (S-uni) ideal form called "soft union (S-uni) tri-bi-ideal”. The main aim of this study is to obtain the relations between S-uni tri-bi-ideals and other certain types of S-uni ideals of a semigroup. Our results show that every S-uni tri-bi-ideal of a band is an S-uni subsemigroup. Moreover, an S-uni tri-bi-ideal is a generalization of an S-uni ideal, interior ideal, bi-ideal, quasi-ideal, weak-interior ideal, bi-interior ideal and bi-quasi ideal, however in order to satisfy the converses, the semigroup should have specific conditions. We also demonstrate that the S-uni quasi-interior ideal of a left or right simple semigroup is an S-uni tri-bi-ideal, nevertheless the converse holds for the zero semigroup. Furthermore, the S-uni bi-quasi-interior ideal of a commutative semigroup is an S-uni tri-bi-ideal, however, for the converse to hold the semigroup must be a band. We have shown that an S-uni tri-ideal coincides with an S-uni tri-bi-ideal of a band, and every S-uni tri-bi-ideal of a group is an S-uni tri-ideal. We also obtain a relation between tri-bi-ideal and its soft characteristic function, enabling us to get the relation between semigroup and soft set theory. Furthermore, we present conceptual characterizations and analysis of the new concept in terms of the soft set operations, the soft (anti/inverse) image, supporting our assertions with illuminating 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 | 0.002 | 0.001 |
| Science and technology studies | 0.002 | 0.004 |
| Scholarly communication | 0.002 | 0.005 |
| Open science | 0.000 | 0.003 |
| Research integrity | 0.000 | 0.002 |
| Insufficient payload (model declined to judge) | 0.005 | 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".