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

Komorníková, Magda; Mesiar, Radko

Fuzzy Sets Syst. 175, No. 1, 48-56 (2011).

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.

Reviewer: Roberto Ghiselli Ricci (Reggio Emilia)

MSC:

06A11

Algebraic aspects of posets

Keywords:

averaging aggregation; disjunctive aggregation; conjunctive aggregation; tolerant aggregation

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