Language of General Fuzzy Recognizer

In this note first by considering the notion of general fuzzy automata (for simplicity GFA), we define the notions of direct product, restricted direct product and join of two GFA. Also, we introduce some operations on (Fuzzy) sets and then prove some related theorems. Finally we construct the general fuzzy recognizers and recognizable sets and give the notion of (trim) reversal of a given GFA. In particular, we define the notion of the language of a given general fuzzy $\Sigma$-recognizer and we show that the language of direct product of two $\Sigma$-recognizer is equal to direct product of their languages.