Some Remarks About Polynomial Aggregation Functions

There exist a great quantity of aggregation functions at disposal to be used in different applications. The choice of one of them over the others in each case depends on many factors. In particular, in order to have an easier implementation, the selected aggregation is required to have an expression as simple as possible. In this line, aggregation functions given by polynomial expressions were investigated in [22]. In this paper we continue this investigation focussing on binary aggregation functions given by polynomial expressions only in a particular sub-domain of the unit square. Specifically, splitting the unit square by using the classical negation, the aggregation function is given by a polynomial of degree one or two in one of the sub-domains and by 0 (or 1) in the other sub-domain. This is done not only in general, but also requiring some additional properties like idempotency, commutativity, associativity, neutral (or absorbing) element and so on, leading to some families of binary polynomial aggregation functions with a non-trivial 0 (or 1) region.

[1]  Michal Baczynski,et al.  Fuzzy Implications: Past, Present, and Future , 2015, Handbook of Computational Intelligence.

[2]  R. Mesiar,et al.  Conjunctors and their Residual Implicators: Characterizations and Construction Methods , 2007 .

[3]  R. Nelsen.  An Introduction to Copulas , 1998 .

[4]  Sebastià Massanet,et al.  On rational fuzzy implication functions , 2016, 2016 IEEE International Conference on Fuzzy Systems (FUZZ-IEEE).

[5]  Joan Torrens,et al.  A survey on the existing classes of uninorms , 2015, J. Intell. Fuzzy Syst..

[6]  János Fodor,et al.  On Rational Uninorms , 2000 .

[7]  R. Mesiar,et al.  Aggregation operators: new trends and applications , 2002 .

[8]  Vicenç Torra,et al.  Modeling Decisions: Information Fusion and Aggregation Operators (Cognitive Technologies) , 2006 .

[9]  Sebastià Massanet,et al.  On (OP)-polynomial implications , 2015, IFSA-EUSFLAT.

[10]  Michal Baczynski,et al.  Fuzzy Implications , 2008, Studies in Fuzziness and Soft Computing.

[11]  Etienne Kerre,et al.  Fuzzy techniques in image processing , 2000 .

[12]  Bernard De Baets,et al.  The functional equations of Frank and Alsina for uninorms and nullnorms , 2001, Fuzzy Sets Syst..

[13]  Sebastià Massanet,et al.  On Fuzzy Polynomial Implications , 2014, IPMU.

[14]  Ronald R. Yager,et al.  Uninorm aggregation operators , 1996, Fuzzy Sets Syst..

[15]  Humberto Bustince,et al.  A review of the relationships between implication, negation and aggregation functions from the point of view of material implication , 2016, Inf. Sci..

[16]  Ronald R. Yager,et al.  Structure of Uninorms , 1997, Int. J. Uncertain. Fuzziness Knowl. Based Syst..

[17]  Joan Torrens,et al.  Aggregation functions given by polynomial functions , 2017, 2017 IEEE International Conference on Fuzzy Systems (FUZZ-IEEE).

[18]  M. J. Frank,et al.  Associative Functions: Triangular Norms And Copulas , 2006 .

[19]  Joan Torrens,et al.  A Survey on Fuzzy Implication Functions , 2007, IEEE Transactions on Fuzzy Systems.

[20]  Francesc Esteva,et al.  Review of Triangular norms by E. P. Klement, R. Mesiar and E. Pap. Kluwer Academic Publishers , 2003 .

[21]  Joan Torrens,et al.  t-Operators , 1999, Int. J. Uncertain. Fuzziness Knowl. Based Syst..

[22]  Gleb Beliakov,et al.  Aggregation Functions: A Guide for Practitioners , 2007, Studies in Fuzziness and Soft Computing.

[23]  Joan Torrens,et al.  The non-contradiction principle related to natural negations of fuzzy implication functions , 2019, Fuzzy Sets Syst..