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A set-theoretic abstraction of some deep ideas from lattice theory is presented and discussed. By making use of this abstraction, many results from seemingly disparate disciplines can be examined, proved, and subtle relationships can be discovered among them. Typical applications might involve decision theory when presented with evidence from sources that yield conflicting optimal advice, insights into the internal structure of a finite lattice, and the nature of homomorphic images of a finite lattice. Some needed historical background is provided. (Presented in conjunction with the volume dedicated to the 70th Birthday celebration of Professor Boris Mirkin.) In particular, there is a connection to some early work of Mirkin (On the problem of reconciling partitions. In: Quantitative Sociology, International Perspectives on Mathematical and Statistical Modelling, pp. 441–449. Academic, New York, 1975).

A study is made of Boolean product representations of bounded lattices over the Stone space of their centres. Special emphasis is placed on relating topological properties such as clopen or regular open equalizers to their equivalent lattice theoretic counterparts. Results are also presented connecting various properties of a lattice with properties of its individual stalks.

In ordinal models for cluster analysis, it has proved useful to study the effect of the action of dissimilarity coefficients by certain isotone mappings. The current work places these results in a graph-theoretic setting, and shows how the resulting theory extends the earlier results obtained in the ordinal models.

Neutral consensus methods have been studied in detail by a number of authors in specific contexts. This paper introduces a general mathematical model that allows for a unified approach to their study via the notion of stability families. Special attention is paid to conditions that produce abstract versions of Arrow's Theorem.

A study is made of the exact nature of a class of cluster methods that are suitable for use with data having only ordinal significance.

An n-median semilattice (n≥3) is a meet-semilattice such that (i) every principal ideal is a distributive lattice and (ii) any n-element set of elements is bounded above whenever each of its (n-1)-element subsets has an upper bound. A 3-median semilattice is thus a median semilattice in the classical sense. In this note we demonstrate how the characteristic features of median semilattices carry over to the more general case of n-median semilattices.

When dissimilarities are measured in a space other than the reals, it is argued that previous models for cluster analysis are not adequate. Possible new models will be explored. It is also shown that formal concept analysis may be viewed as a special case of a Boolean dissimilarity coefficient. A persistent underlying theme involves generalized notions of adjoints of order preserving mappings between posets.

We use a set theoretic approach to consensus by viewing an object as a set of smaller pieces called "bricks". A consensus function is neutral if there exists a family D of sets such that a brick s is in the output of a profile if and only if the set of positions with objects that contain s belongs to D. We give sufficient set theoretic conditions for D to be a lattice filter and, in the case of a finite lattice, these conditions turn out to be necessary. Our final result, which involves a finite distributive join semilattice, provides necessary and sufficient conditions for D to be an ultrafilter. Nous abordons le problème du concsensus par une voie ensembliste, en considérant un objet comme un assemblage de "briques" élémentaires. Une fonction de consensus est neutre s'il existe une famille D d'ensembles telle qu'une brique s appartient au consensus d'un profil si et seulement si l'ensemble des coordonnées des objets contenant s appartient à D. Nous donnons des conditions suffisantes pour que D soit une filtre de treillis. Dans le cas d'un treillis fini, ces conditions s'avèrent être aussi suffisantes. Notre résultat final porte sur le cas d'un sup-demis-treillis distributif fini, dans lequel nous donnons des conditions nécessaires et suffisantes pour que D soit un ultrafiltre.

Several authors have investigated consensus functions from the viewpoint of stability families. It will be shown that the stability family approach extends and enriches the ordered set approach to consensus theory. Our investigation will center on the notions of sup and inf-irreducible elements of a finite ordered set. We will see that irreducible elements give rise to natural internal stability families for a given ordered set X. We will investigate the relationship between these internal stability families and abstract stability families. Special attention will be paid to the study of different types of neutrality, thus extending earlier work of B. Monjardet. We will be particularly interested in the change in the available neutral consensus functions on an ordered set when a given internal stability family is changed.

The conference was held October 14-17, 1993. There were a total of 59 participants from 10 different countries. The focus of the conference centered on the following topics: (1) What is the nature of ordinal data?; (2) How does ordinal data arise and what are its uses?; (3) What techniques have been used in the past to help interpret various types of ordinal data?; and (4) What directions should be taken by future research?

Order theoretic and combinatorial properties of the semilattice of weak orders on a set are developed. In the case of a finite set, an order theoretic characterization of this semilattice is obtained.

The central purpose of this paper is to put together the basic ideas of two separate theories - the theory of ordinal clustering, as developed by Janowitz et al., and the theory of probabilistic metric spaces, as developed by Schweizer et al. The principal result is a new theory of clustering, called percentile clustering, in which the clustering is based, not on some average or other typical value of the data, but directly on the distributed data itself. A secondary outgrowth is a generalized theory of ordinal clustering. The paper begins with a brief but essentially self-contained exposition of the main ideas of the two theories mentioned above. It then combines them to lay the foundations of the theory of percentile clustering. It goes on to develop a number of specific algorithms with a small, artificial data set. The paper concludes by applying the new cluster methods to two concrete examples. The first of these is a data set concerning combat deaths in the Vietnam War; the second is a data set supplied by N. Creel and dealing with the classification of species of gibbons. In both instances results obtained with various standard clustering techniques are also presented.

The transitive closure of a complete quasi-transitive relation is a weak order. This fact is used to define various social welfare functions that are generalizations of the Pareto extension rule as well as certain median consensus rules. The resulting functions are axiomatically characterized, and some of their properties are developed. A relation is shown between these functions and a generalized form of compatibility.

We show that a reducible continuous geometry can be represented as the continuous sections of a bundle of irreducible continuous geometries. We relate this bundle representation to the Pierce sheaf of the continuous geometry and to the subdirect product representation developed by Maeda.