Compactness of fuzzy logics

Compactness is an important property of classical logic. It states that simultaneous satisfiability of an infinite set of formulas is equivalent to the satisfiability of all its finite subsets. In fuzzy logics, we have different degrees of satisfiability, hence the questions of compactness become more complicated. Here we give an overview of the recent results on compactness and we extend them to various fuzzy logics.