Prog Artif Intell (2016) 5:1–14 DOI 10.1007/s13748-015-0069-6 REGULAR PAPER On two construction methods of copulas from fuzzy implication functions Sebastia Massanet 1 · Daniel Ruiz-Aguilera 1 · Joan Torrens 1 Received: 20 November 2015 / Accepted: 23 November 2015 / Published online: 14 December 2015 © Springer-Verlag Berlin Heidelberg 2015 Abstract Copulas have been deeply investigated because of their applications in many fields. From the theoretical point of view, a key point in this research lies in the search of new construction methods of parametrized families of cop- ulas. This paper presents some construction methods based on fuzzy implication functions by reversing the construc- tion methods of fuzzy implication functions from copulas presented by P. Grzegorzewski in some recent papers. Specif- ically, the PSI and SSI-construction methods of copulas are proposed which provide copulas from a given fuzzy implica- tion function. In addition, the analysis of these construction methods of copulas lead to the characterization of the inter- section of the probabilistic S and survival S-implications with ( S, N ) and R-implications. Keywords Fuzzy implication function · Copula · t-conorm · Survival copula 1 Introduction Copulas [27] are a special kind of aggregation functions that come from probability theory and statistics and have important applications not only in these fields, but also in economics mainly. The study of copulas has grown in last B Sebastia Massanet s.massanet@uib.es Daniel Ruiz-Aguilera daniel.ruiz@uib.es Joan Torrens jts224@uib.es 1 Department of Mathematics and Computer Science, University of the Balearic Islands, Crta. Valldemossa km. 7.5, Palma, Spain decades and one of the main topics in this sense lies in the research of different construction methods of copulas or parametrized families of copulas. On the other hand, fuzzy implication functions [4, 7] are a kind of logical operators that come from fuzzy sets and fuzzy logic and they have a lot of applications that vary from approximate reasoning and fuzzy control [17, 22] to fuzzy subset-hood measures and image processing [9, 10], specially in fuzzy mathematical morphol- ogy [16]. In spite of their very different origins, both kinds of opera- tors are highly related between them. Indeed, there are many published works dealing with the relationship among aggre- gation functions in general (including copulas) and fuzzy implication functions (see [4, 8, 23, 28] and references there- in). In particular, many authors have investigated methods to construct fuzzy implication functions from aggregation functions and vice versa. A well-known example of these methods comes from the family of residual implications or R-implications in short. Indeed, one of the earliest meth- ods for obtaining implications was from conjunctions as their residuals, and many different classes of conjunctions have been used in this line. Mainly, left-continuous t-norms [23], left-continuous conjunctive uninorms (leading to the so- called RU -implications) [1, 12], but also semi-copulas and copulas [14] and many other types of conjunctive aggrega- tion functions (see [26] and references therein). In the major part of these cases, it is also possible to construct the initial conjunction from the corresponding residual implication. Another construction method comes from the material implications. In this case, given a negation N , it is possi- ble to construct an implication function from a t-conorm S, obtaining the well-known ( S, N )-implications, and vice versa [4, 5]. Again, one can use a disjunctive uninorm instead of a t-conorm (obtaining the so-called (U, N )-implications) [6], and also many other kinds of disjunctive aggregation 123