Galois Connections in Axiomatic Aggregation

Abstract

We investigate the relations between, on the one hand, Galois connections and the related types of maps and, on the other hand, the axiomatic Arrowian approach for the aggregation (or consensus) problem in lattices. In the latter one wants to ”aggregate” n-tuples (n ≥ 2) of elements of a lattice L into an element of this lattice representing their ”consensus”, subject to satisfying some desirable properties. The main axiom is a generalization of Arrow’s [1] independence. The results consist in the characterization of convenient aggregation functions, and especially in impossibility ones when axioms turn to be incompatible. For the many applications of this theory in the domains of social choice or cluster analysis, see, e.g., the book of Day and McMorris [4]. Basic characterizations of Arrowian aggregation functions according to a specific typology of finite lattices are given by Monjardet [10]. They are extended to lattices of Galois maps (or polarized ones, that is maps appearing in Galois connections), then particularized to fuzzy preorders and hierarchical classifications, in Leclerc [7]. A unified presentation is given in Leclerc and Monjardet [8].

An FCA-related representation of Galois maps between two fixed lattices is given in Domenach and Leclerc [5] with the introduction of the so-called ”biclosed” relations. As pointed out in the unifying paper of Ganter [6], the notion of biclosed relations is related to several others in the literature. The first part of the presentation will be devoted to Arrowian aggregation of biclosed relations.

In the second part, we present another relation between Aggregation theory and residuated/residual maps (those appearing in Residuation Theory [2]), which corresponds to ”covariant” Galois connections. Chambers and Miller [3] and Leclerc and Monjardet [9] have recently pointed out that, in a significant class of atomistic lattices, an aggregation function is a meet-projection if and only if it is a residual mapping.