Artificial intelligent Global Online Learning (GOL) theory by generalized n-ary fuzzy relation

Abstract Following the devastating COVID pandemic, a new global strategy is required to switch from traditional education to blended or online learning method. Nevertheless, there is no adamant theoretical base available for such an important transition or similar situations in the future. On the other hand, educational systems encounter uncertainty as an integral part of multilayered teaching routes. To analyze the interactions among interconnected entities, soft computing methodologies can serve as an efficient tool to manage such systems with uncertain information through incorporating artificial intelligence (AI) techniques for assessing students performances. Nevertheless, the classical binary fuzzy relation and other existing theoretical models are not capable of explaining/configuring uncertain-based datum for multiplex correlations. To fill these gaps, the present study establishes a neoteric AI-base “Global Online Learning (GOL) theory” using the newly developed n -ary relation and n -ary fuzzy relation as the generalization of classical and binary fuzzy relations. Through the enhanced mathematical concepts and intelligent soft computing techniques, the convoluted multilayer relationships of entities can be punctiliously assessed for different values of n. Furthermore, a network-based perspective is proposed as a promising systematic model when systems are imperfect and prone to uncertainty. In the provided graphical context, the n -ary relation represents the hypergraph pattern, while the n -ary fuzzy relation refers to the generalized fuzzy hypergraph model. Fundamental characteristics of n -ary fuzzy relation, including reflexive, symmetric, transitive, composition, t -cut, support and Cartesian product, are systematically provided to extract mathematical interrelated expressions, as well as parametric connection between t -cut and Cartesian product. Based on the n -ary fuzzy relation, the n -ary fuzzy hyperoperation “ $$\circ _{\rho }$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mo>∘</mml:mo> <mml:mi>ρ</mml:mi> </mml:msub> </mml:math> ” is assigned to construct fuzzy hyperalgebra as the extension of classical algebra with illustrative examples. The relationships between fuzzy hyperalgebra and hyperalgebra are investigated through the notation of $$(\circ _{\rho })_{t}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mrow> <mml:mo>(</mml:mo> <mml:msub> <mml:mo>∘</mml:mo> <mml:mi>ρ</mml:mi> </mml:msub> <mml:mo>)</mml:mo> </mml:mrow> <mml:mi>t</mml:mi> </mml:msub> </mml:math> for $$t\in (0,1].$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>t</mml:mi> <mml:mo>∈</mml:mo> <mml:mo>(</mml:mo> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:mn>1</mml:mn> <mml:mo>]</mml:mo> <mml:mo>.</mml:mo> </mml:mrow> </mml:math> With the introduced t -cut methodology, the corresponding hypergraph is derived to simplify the analysis of educational information. The AI-base GOL theory provides a solid gadget for learning data management, e.g., the grading evaluation of online assessments, where the evaluation of components is accomplished on real data in terms of fuzzy n -ary relation, t -cut and support through a graphical attitude. The results indicate that the AI-base GOL theory is a robust platform to meticulously manage and control uncertain-based intercorrelated information. This platform can be converted into a coding gadget for artificial intelligent educational online mega-systems.