Let (H, α) be a monoidal Hom-bialgebra and (B, β) be a left (H, α)-Hom-module algebra and also a left (H, α)-Hom-comodule coalgebra. Then in this paper, we first introduce the notion of a Hom-smash coproduct, which is a monoidal Hom-coalgebra. Second, we find sufficient and necessary conditions for the Hom-smash product algebra structure and the Hom-smash coproduct coalgebra structure on B ⊗ H to afford B ⊗ H a monoidal Hom-bialgebra structure, generalizing the well-known Radford's biproduct, where the conditions are equivalent to that (B, β) is a bialgebra in the category of Hom-Yetter-Drinfeld modules $^{H} _{H}\mathcal {HYD}$HYDHH. Finally, we illustrate the category of Hom-Yetter-Drinfeld modules $^{H} _{H}\mathcal {HYD}$HYDHH and prove that the category $^{H} _{H}\mathcal {HYD}$HYDHH is a braided monoidal category.

No abstract

Information is considered to be the major part of an organization. With the enhancement of technology, the knowledge level is increasing with the passage of time. This increase of information is in volume, velocity, and variety. Extracting meaningful insights is the dire need of an individual from such information and knowledge. Visualization is a key tool and has become one of the most significant platforms for interpreting, extracting, and communicating information. The current study is an endeavour toward data modelling and user knowledge by using a rough set approach for extracting meaningful insights. The technique has used different rough set algorithms such as K-nearest neighbours (KNN), decision rules (DR), decomposition tree (DT), and local transfer function classifier (LTF-C) for an experimental setup. The approach has found its accuracy for the optimal use of data modelling and user knowledge. The experimental setup of the proposed method is validated by using the dataset available in the UCI web repository. Results of the proposed study show that the model is effective and efficient with an accuracy of 96% for KNN, 87% for decision rules, 91% for decision trees, 85.04% for cross validation architecture, and 94.3% for local transfer function classifier. The validity of the proposed classification algorithms is tested using different performance metrics such as F-score, precision, accuracy, recall, specificity, and misclassification rates. For all these performance metrics, the KNN classifier outperformed, and this high performance shows the applicability of the KNN classifier in the proposed problem.

The notion of a crossed product with a Hopf algebroid was introduced by Böhm and Brzeziński [12]. It is well-known that weak Hopf algebras (see Böhm et al. [10])is an example of Hopf algebroids. Then we get the definition and some property of a weak crossed product A# H over an algebra A and a weak Hopf algebra H. Next we give a Maschke-type theorem for the weak crossed product over a semisimple weak Hopf algebra H. Furthermore, we obtain an analogue of the Nikshych's duality theorem for weak crossed products. Finally, using this duality theorem, we prove that the global dimension of A equals to the global dimension of A# H if H and H * are both semisimple.