Theories of fuzzy sets and type-2 fuzzy sets are powerful mathematical tools for modeling various types of uncertainty. In this paper we introduce the concept of type-2 fuzzy finite state automata and discuss the algebraic study of type-2 fuzzy finite state automata, i.e., to introduce the concept of homomorphisms between two type-2 fuzzy finite state automata, to associate a type-2 fuzzy transformation semigroup with a type-2 fuzzy finite state automata. Finally, we discuss several product of type-2 fuzzy finite state automata and shown that these product is a categorical product.

In this note, we show that, in the paper [Zhong Y, Yan CH. Intuitionistic L-fuzzy rough sets, intuitionistic L-fuzzy preorders and intuitionistic L-fuzzy topologies. Fuzzy Inf Eng. 2016;8:255-279.], the conclusion regarding Proposition 3.3 of [Tiwari SP, Srivastava AK. Fuzzy rough sets, fuzzy preorders and fuzzy topologies. Fuzzy Sets Syst. 2013;210:63-68.] is not correct.