A Fuzzy Algebraic System Based on the Theory of Falling Shadows

Based on the concept of a falling shadow, a theoretical approach of the fuzzy algebraic system is established. A fuzzy subalgebraic system of an algebraic system is considered as a falling shadow of the “cloud of the subalgebraic systems.” When an algebraic system is a ring, fuzzy subrings (fuzzy ideas, fuzzy prime ideas) of a ring are seen as falling shadows of “the cloud of subrings” (“cloud of ideas,” “cloud of prime ideas”). We shall show that fuzzy algebraic systems defined int-norms are consequences of our definition under certain conditions.