Functional System of BL-Algebra with Ininite Formulas
The class of functions which can be represented by formulas of a standard BL-algebra whose basic operation is a t-norm is fully characterized. For the purpose of representation, we consider formulas (including special formulas of an infinite length) which are constructed from variables and constants from [0, 1] with the help of operations from a BL-algebra. We prove that the class of functions which are represented by such formulas coincides with the class of extensional functions. The proof is based on the uniform representation by the generalized normal (disjunctive and conjunctive) forms. Moreover, the class of these functions coincides with the class of Lipschitz continuous functions in the case of BL-algebra whose t-norm has a continuous additive generator.