Extensions of Discrete Copulas to Sparse Copulas

In this article, we answer positively an open question posed in [G. Mayor, J. Suner, and J. Torrens, Copula-like operations on finite settings, IEEE Trans. Fuzzy Syst., vol. 13, no. 4, pp. 468-477, Aug. 2005] concerning the extensions of discrete copulas to shuffles of Min. Moreover, we also use the extension of discrete copulas to sparse copulas, associated with discrete copulas, of idempotent copulas, in particular, the copulaM, in order to answer the open question in an alternative way.

Automated summary

In this article, we affirmatively answer an open question regarding the extensions of discrete copulas to shuffles of Min, as posed in [G. Mayor, J. Suner, and J. Torrens, Copula-like operations on finite settings, IEEE Trans. Fuzzy Syst., vol. 13, no. 4, pp. 468-477, Aug. 2005]. Additionally, we utilize the extension of discrete copulas to sparse copulas associated with idempotent copulas, particularly the copulaM, in an alternative manner to address the open question.