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 in t-norms are consequences of our definition under certain conditions