A characterization of tribes with respect to the Łukasiewicz $t$-norm

summary:We give a complete characterization of tribes with respect to the Łukasiewicz $t$-norm, i. e., of systems of fuzzy sets which are closed with respect to the complement of fuzzy sets and with respect to countably many applications of the Łukasiewicz $t$-norm. We also characterize all operations with respect to which all such tribes are closed. This generalizes the characterizations obtained so far for other fundamental $t$-norms, e. g., for the product $t$-norm