Regularity and Green's Relations for Generalized Semigroups of Transformations with Invariant Set
Bibliographic record
Abstract
Let ${\cal T}_X$ be the full transformation semigroup on a set $X$.For $Y\subseteq X$, the semigroup $S(X,Y) =\{ f\in {\cal T}_X: f(Y)\subseteq Y\}$ is a subsemigroup of ${\cal T}_ X $. Fix an element $\theta\in S(X,Y)$ and for $f,g\in S(X,Y)$, define a new operation $*$ on $S(X,Y)$ by $f* g=f\theta g$ where $f\theta g$ denotes the produce of $g,\theta$ and $f$ in the original sense. Under this operation, the semigroup $S(X,Y)$ forms a semigroup which is called generalized semigroup of $S(X,Y)$ with the sandwich function $\theta$ and denoted by $S(X,Y,*_\theta)$. In this paper we first characterize the regular elements and then describe Green's relations for the semigroup $S(X,Y,*_\theta)$.
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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 | 0.001 | 0.003 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.002 | 0.003 |
| Scholarly communication | 0.001 | 0.003 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.001 | 0.001 |
| Insufficient payload (model declined to judge) | 0.003 | 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 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".