L-fuzzy algebraic substructure

This article aims to provide a method for defining L-fuzzy algebraic substructures on general algebras. Concretely, the properties of L-fuzzy sets are first reviewed, and their representations are then provided. Next, algebraic substructures are generalised as the closure systems on the power set of the algebra, and the properties of the prime and maximal elements in the above closure system are investigated. Based on these facts, L-fuzzy algebraic substructures with respect to the closure system are defined and studied. Two equivalence characterisations of the sup property of the ordered set L are provided using L-fuzzy substructures. Similarly, some properties of L-fuzzy prime and maximal substructures with respect to the closure system are discussed. Finally, to demonstrate the broad applicability of the theory of L-fuzzy algebraic substructures, the theory is applied to some specific algebraic structures, such as groups and pseudo MV-algebras.