The notions of a (limited) T & F-ideal and a T & F--subalgebra are introduced, and various properties are investigated. The relationship between a T & F-subalgebra and a T & F-ideal is established. Conditions for a T & F-structure (resp., T & F-subalgebra) to be a T & F-ideal are discussed. An example is provided to confirm that in a BCI-algebra, a T & F-ideal cannot be a T & F-subalgebra in general, and then conditions for a T & F-ideal to be a T & F-subalgebra in a BCI-algebra are considered. The relationship between a T & F-ideal and a T & F--subalgebra in a BCK-algebra with condition (S) is discussed. Conditions for a T & F-structure over a BCK-algebra with condition (S) to be a T & F-ideal are provided. Using the notion of T & F-level sets, a characterization of a T & F-ideal is established. A novel T & F-ideal is constructed with an existing T & F-ideal using a characterization of a T & F-ideal.