Fully ordered semigroups and their applications

E. Ya. Gabovich

Trudy Matematicheskogo Instituta imeni V.A. Steklova, Tome 31 (1976) no. 1, pp. 147-216

Résumé

This is a survey of the theory of fully ordered (f.o.) semigroups as it stands at the present time. In addition to three chapters on the major trends of research into the theory of f.o. semigroups, namely, ‘Orderability conditions’, ‘Constructions’ and ‘Structure theory’, we include a chapter on the applications of the theory in other areas of algebra, in abstract measurement theory, and in discrete mathematical programming. In Chapter I we consider the orderability of the free semigroups in several varieties and the representability of f.o. semigroups as o-epimorphic images of ordered free semigroups. We also examine the question of how many orderings there are on an f.o. semigroup, and give orderability criteria for certain classes of semigroups. In Chapter II we consider the question of the orderability of bands of f.o. semigroups, and of lexicographic and free products of f.o. semigroups. We also study the classes of c-simple and o-simple f.o. semigroups, and representations of f.o. semigroups. In Chapter III we investigate the partitioning of an f.o. semigroup into Archimedean components. We give a description of f.o. idempotent semigroups and of various other classes of f.o. semigroups, and we examine the structure of convex subsemigroups of f.o. semigroups. We draw parallels with other areas in the theory of f.o. algebraic systems. Twenty-two problems are incorporated into the text.