##
**Aggregation functions on bounded partially ordered sets and their classification.**
*(English)*
Zbl 1253.06004

The main purpose of this paper is to provide a classification of the aggregation functions defined on a partial ordered set (poset), trying to generalize the existing, sharp classification when the domain and the range of the functions is a real interval, that is, conjunctive, disjunctive, averaging and the remaining ones, denoted by \((\mathcal C, \mathcal D, \mathcal P, \mathcal R)\).

In the third section, after the introduction and some basic notions on this topic (particularly, the definition of aggregation function on a poset \(P\), that is, a mapping \(A:P^n\to P\) which satisfies some border conditions and is increasing, i.e. \(A(\mathbf{x})\leq A(\mathbf{y})\) whenever \(\mathbf{x}\leq\mathbf{y}\)), we may read the main idea, simply based upon a pair of functions given by \(g_A(\mathbf{x})= \text{card}\{i\mid x_i\geq A(\mathbf{x})\}\) and \(s_A(\mathbf{x})= \text{card}\{i\mid x_i\leq A(\mathbf{x})\}\), which count how many projections of the generic vector \(\mathbf{x}\) are respectively greater or lower than \(A(\mathbf{x})\). Then, passing to the infimum of \(g_A\) and \(s_A\), given by \(\gamma(A)\) and \(\sigma(A)\) respectively, it is not difficult to see that \(\{\mathcal C_i\}_{i=1,\dots,n}\), where \(\mathcal C_i=\gamma^{-1}(\{i\})\), is a partition of the whole family of the aggregation functions on \(P\) (and the same may be done starting with \(\sigma\)). So, for instance, an aggregation function from the class \(\mathcal C_n\) satisfies \(A(\mathbf{x})\leq x_i\) for all \(i=1,\dots,n\), and it is called strongly conjunctive, while the aggregation functions from \(\mathcal C_w=\bigcup_{i=1}^{n-1}\mathcal C_i\) are called weakly conjunctive (and analogously for strongly and weakly disjunctive).

In the fourth section, the definition of an aggregation function strongly or weakly averaging is provided, according to the fact that \(A\) is both weakly conjunctive and disjunctive or either weakly conjunctive or disjunctive. In the fifth section, in the case of a chain, the authors show that their classification clearly reduces to the well-known case. Finally, some hints at product posets are added.

In the third section, after the introduction and some basic notions on this topic (particularly, the definition of aggregation function on a poset \(P\), that is, a mapping \(A:P^n\to P\) which satisfies some border conditions and is increasing, i.e. \(A(\mathbf{x})\leq A(\mathbf{y})\) whenever \(\mathbf{x}\leq\mathbf{y}\)), we may read the main idea, simply based upon a pair of functions given by \(g_A(\mathbf{x})= \text{card}\{i\mid x_i\geq A(\mathbf{x})\}\) and \(s_A(\mathbf{x})= \text{card}\{i\mid x_i\leq A(\mathbf{x})\}\), which count how many projections of the generic vector \(\mathbf{x}\) are respectively greater or lower than \(A(\mathbf{x})\). Then, passing to the infimum of \(g_A\) and \(s_A\), given by \(\gamma(A)\) and \(\sigma(A)\) respectively, it is not difficult to see that \(\{\mathcal C_i\}_{i=1,\dots,n}\), where \(\mathcal C_i=\gamma^{-1}(\{i\})\), is a partition of the whole family of the aggregation functions on \(P\) (and the same may be done starting with \(\sigma\)). So, for instance, an aggregation function from the class \(\mathcal C_n\) satisfies \(A(\mathbf{x})\leq x_i\) for all \(i=1,\dots,n\), and it is called strongly conjunctive, while the aggregation functions from \(\mathcal C_w=\bigcup_{i=1}^{n-1}\mathcal C_i\) are called weakly conjunctive (and analogously for strongly and weakly disjunctive).

In the fourth section, the definition of an aggregation function strongly or weakly averaging is provided, according to the fact that \(A\) is both weakly conjunctive and disjunctive or either weakly conjunctive or disjunctive. In the fifth section, in the case of a chain, the authors show that their classification clearly reduces to the well-known case. Finally, some hints at product posets are added.

Reviewer: Roberto Ghiselli Ricci (Reggio Emilia)

### MSC:

06A11 | Algebraic aspects of posets |

### Keywords:

averaging aggregation; disjunctive aggregation; conjunctive aggregation; tolerant aggregation
PDF
BibTeX
XML
Cite

\textit{M. Komorníková} and \textit{R. Mesiar}, Fuzzy Sets Syst. 175, No. 1, 48--56 (2011; Zbl 1253.06004)

Full Text:
DOI

### References:

[1] | Beliakov, G.; Pradera, A.; Calvo, T., Aggregation functions: A guide for practitioners, (2007), Springer-Verlag Berlin, Heidelberg · Zbl 1123.68124 |

[2] | G. Birkhoff, Lattice Theory, American Mathematical Society Colloquium Publications, third ed., vol. XXV, American Mathematical Society, Providence, RI, 1967. |

[3] | Calvo, T.; Kolesárová, A.; Komorníková, M.; Mesiar, R., Aggregation operators: properties, classes and construction methods, (), 1-104 · Zbl 1039.03015 |

[4] | De Baets, B.; Mesiar, R., Triangular norms on product lattices, Fuzzy sets and systems, 104, 61-76, (1999) · Zbl 0935.03060 |

[5] | De Cooman, G.; Kerre, E.E., Order norms on bounded partially ordered sets, The journal of fuzzy mathematics, 2, 281-310, (1994) · Zbl 0814.04005 |

[6] | Demirci, M., Aggregation operators on partially ordered sets and their categorical foundations, Kybernetika, 42, 261-277, (2006) · Zbl 1249.03091 |

[7] | Deschrijver, G., Arithmetic operators in interval-valued fuzzy set theory, Information sciences, 177, 14, 2906-2924, (2007) · Zbl 1120.03033 |

[8] | Deschrijver, G., Characterizations of (weakly) Archimedean t-norms in interval-valued fuzzy set theory, Fuzzy sets and systems, 160, 6, 778-801, (2009) · Zbl 1181.03052 |

[9] | Deschrijver, G., Generalized arithmetic operators and their relationship to t-norms in interval-valued fuzzy set theory, Fuzzy sets and systems, 160, 21, 3080-3102, (2009) · Zbl 1183.03050 |

[10] | Deschrijver, G.; Kerre, E.E., Uninorms in \(L^*\)-fuzzy set theory, Fuzzy sets and systems, 148, 243-262, (2004) · Zbl 1058.03056 |

[11] | Deschrijver, G.; Kerre, E., Aggregation operators in interval-valued fuzzy and Atanassov’s intuitionistic fuzzy set theory, Studies in fuzziness and soft computing, 220, 183-203, (2008) · Zbl 1201.68118 |

[12] | Dubois, D.; Prade, H., On the use of aggregation operations in information fusion processes, Fuzzy sets and systems, 142, 143-161, (2004) · Zbl 1091.68107 |

[13] | Grabisch, M.; Marichal, J.-L.; Mesiar, R.; Pap, E., Aggregation functions, (2009), Cambridge University Press Cambridge |

[14] | Jenei, S.; De Baets, B., On the direct decomposability of t-norms on product lattices, Fuzzy sets and systems, 139, 3, 699-707, (2003) · Zbl 1032.03022 |

[15] | Karacal, F.; Khadjiev, D., \(\vee - \operatorname{distributive}\) and infinitely \(\vee - \operatorname{distributive}\) t-norms on complete lattices, Fuzzy sets and systems, 151, 2, 341-352, (2005) · Zbl 1062.06007 |

[16] | Marichal, J.-L., K-intolerant capacities and Choquet integrals, European journal of operational research, 177, 3, 1453-1468, (2007) · Zbl 1110.90053 |

[17] | R. Mesiar, M. Komorníková, Classification of aggregation functions on bounded partially ordered sets, in: Proceedings of the SISY’ 2010, Subotica, September 10-11, 2010, pp. 13-16. |

[18] | Saminger-Platz, S.; Klement, E.P.; Mesiar, R., On extensions of triangular norms on bounded lattices, Indagationes mathematicae, 19, 1, 135-150, (2008) · Zbl 1171.03011 |

[19] | Turksen, I.B., Interval-valued fuzzy sets and ‘compensatory AND’, Fuzzy sets and systems, 51, 3, 295-307, (1992) |

[20] | Zhang, D., Triangular norms on partially ordered sets, Fuzzy sets and systems, 153, 2, 195-209, (2005) · Zbl 1091.03025 |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.