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 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 | 0.000 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.001 | 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 itClassification
machine, unvalidatedMachine predicted; a candidate call from one teacher head, 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".