This paper proposes a way to encode exhaustivity in clefts as a presupposition, something which has been claimed to be adequate, but never successfully implemented. We furthermore show that the facts that prompted the need for such an analysis carry over to identity sentences with definite DPs and propose a way to achieve the same presuppositions for definite DPs.
Büring, D., & Križ, M. (2013). It’s that, and that’s it! Exhaustivity and homogeneity presuppositions in clefts (and definites). Semantics and Pragmatics, 6. https://doi.org/10.3765/sp.6.6