Preface

Summary

The notions of order, classification, and ranking exist in numerous human activities and situations: administrative or social hierarchies, organization charts, scheduling of sports tournaments, precedence, succession or preference orders, agendas, school, audiovisual or webpage rankings, alphabetical and lexicographic orders, etc. It would be endless to enumerate all the situations where orders appear.

It is thus not surprising, considering the development of the use of mathematics in the modeling of multiple phenomena, to find a great number of fields where order mathematics occur. Nevertheless, the latter are relatively recent. Of course, in mathematics, the notion of the order of magnitude has been known for a long time and in the sixteenth century the symbols “<” and “>” appeared for the first time to express “less than” and “greater than.” Yet, the abstract notion of an order defined as a particular type of transitive relation was developed only between 1880 and 1914 by mathematicians and/or logicians such as Peirce, Peano, Schröder, Cantor, Dedekind, Russell, Huntington, Scheffer, and Hausdorff, in the context of the formalization of the “algebra of logic” (that is, Boolean algebra) and also of the creation of set theory (with the study of “order types”). Lattices, which are particular orders since they can be defined algebraically, were also considered as early as the later part of the nineteenth century by Schröder and Dedekind, and then fell into oblivion before arising again during the 1930s thanks to Birkhoff, Öre, and several other eminent mathematicians.