Discrete duality is a type of duality between a class \(\mathrm{Alg}\) of algebras and a class \(\mathrm{Fr}\) of relational systems referred to as frames, where every algebra in \(\mathrm{Alg}\) is embeddable into the complex algebra of its canonical frame, and every frame in \(\mathrm{Fr}\) is embeddable into the canonical frame of its complex algebra. In [the second author and \textit{A. M. Radzikowska}, in: Witnessed years. Essays in honour of Petr Hájek. London: College Publications. 329--344 (2009; Zbl 1225.03093)] and [the second author and \textit{I. Rewitzky}, Fuzzy Sets Syst. 161, No. 9, 1325--1342 (2010; Zbl 1195.03059)] discrete dualities were developed for the class of MTL-algebras (monoidal t-norm based logic algebras). In the present paper, the authors establish that the discrete duality for MTL-algebras extends to \(n\)-potent MTL-algebras (i.e. satisfying \(x^n = x^{n+1}\)). They also consider BL-algebras (algebras of Hajek's basic logic), which form a subvariety of the variety of MTL-algebras, and provide a discrete duality for \(2\)-potent BL-algebras (i.e. satisfying \(x^2 = x^3\).)