Structural Properties of (l,r)- and (r, l)-Derivations in IUP-Algebras
Bibliographic record
Abstract
The notion of derivations in BCI-algebras was initially introduced by Jun and Xin in 2004 [18], providing a foundation for structural analysis in non-classical logics. In this paper, we extend the study of derivations to the framework of IUP-algebras X = (X, ·, 0) by introducing and examining two new types: (l,r)-derivations and (r, l)- derivations. These operators are defined via the binary operation ∧ given by x∧y = (y · x)· x for all x, y ∈ X, which plays a central role in the algebraic structure. We explore fundamental properties of these derivations, analyze their interaction with IUP-substructures, and establish several characterizations. Additionally, we define two special subsets—Kerd(X) (the kernel) and Fixd(X) (the fixed-point set)—associated with a derivation d, and investigate conditions under which they exhibit algebraic regularity. Our results enrich the theory of derivations in IUP-algebras and open new directions for the study of morphism-based operations in non-associative systems.
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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 | 0.006 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.002 | 0.001 |
| Science and technology studies | 0.002 | 0.004 |
| Scholarly communication | 0.002 | 0.005 |
| Open science | 0.001 | 0.002 |
| Research integrity | 0.001 | 0.002 |
| Insufficient payload (model declined to judge) | 0.002 | 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".