The authors study the Brouwer-Zadeh MV-algebras. Like in the MV case [C. C. Chang, “Algebraic analysis of many valued logics”, Trans. Am. Math. Soc. 88, 467-490 (1958; Zbl 0084.00704)] they consider two pairs of conjuction and disjunction: an idempotent pair (\(\vee ,\wedge \)), and a non-idempotent one (\(\odot ,\oplus \)). Like in the BZ case [see G. Cattaneo and G. Marino, Fuzzy Sets Syst. 25, 107-123 (1988; Zbl 0631.06005)] they consider a fuzzy negation \(\neg \) and an intuitionistic-like negation \(\sim \). In this framework they prove the categorical equivalence between different kinds of structures characterizing some many-valued logics. Particularly they investigate Chang’s MV-algebras [R. Cignoli and A. Monteiro, Proc. Japan Acad. 41, 676-680 (1965; Zbl 0168.00602)] and Stonian MV-algebras [L. P. Belluce, J. Math. Anal. Appl. 205, 485-499 (1997; Zbl 0874.06010)].