Soft Intersection Quasi-interior Ideals of Semigroups
Bibliographic record
Abstract
It has been shown that generalizing the ideals of an algebraic structure is both interesting and beneficial for mathematicians. In this context, the concept of quasi-interior (Ԛꟾ) ideal was introduced as a generalization of quasi-ideal and interior ideal of a semigroup. In this paper, we apply this concept to soft set theory and semigroups, introducing a new form of soft intersection (S-int) ideal called the "soft intersection (S-int) quasi-interior (Ԛꟾ) ideal." The main objective of this study is to investigate the relationships between S-int Ԛꟾ ideals and other specific types of S-int ideals in a semigroup. It has been shown that every S-int interior ideal of a semigroup is an S-int Ԛꟾ ideal, and every S-int ideal is an S-int Ԛꟾ ideal. The S-int bi-ideal of a group is an S-int Ԛꟾ ideal, the S-int quasi-ideal of a regular group is an S-int Ԛꟾ ideal, the idempotent S-int Ԛꟾ ideal is an S-int bi-quasi-ideal and an S-int bi-interior ideal. Counterexamples are provided to show that the opposites of these statements are not always valid. We prove that for the converses to hold, the semigroup should be a group or regular, or the S-int Ԛꟾ ideal should be idempotent. Our main theorem, which demonstrates that if a subsemigroup of a semigroup is a Ԛꟾ ideal, then its soft characteristic function is an S-int Ԛꟾ ideal, and vice versa, enables us to establish a connection between semigroup theory and soft set theory. Through this theorem, we illustrate how this concept connects to the existing algebraic structures in classical semigroup theory. Additionally, we offer conceptual characterizations and an analysis of the concept in terms of soft set operations, including soft image and soft inverse image, supporting our claims with specific, informative examples. Furthermore, the connection between a regular semigroup and the structure of S-int Ԛꟾ ideals is established and presented.
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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 | 0.000 |
| 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) | 0.004 | 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".