X PREFACE semigroups with minimal conditions, etc.) and their representation by mappings or by matrices. The second half of Volume 2 treats the theory of congruences and the embedding of semigroups in groups, including a modest account of the active French school of semigroups (which they call "demi-groupes") founded in 1941 by P. Dubreil. In order to keep our book within reasonable bounds, moreover, we have construed the term "algebraic'' in a somewhat narrow sense: the semigroups under consideration are not endowed with any further structure. This has the effect of excluding not only topological semigroups, but ordered semi- groups as well. Fortunately, a good account of lattice-ordered semigroups and groups is to be found in G. BirkhofTs Lattice Theory (Amer. Math. Soc. Colloq. Publ., 1940 revised 1948). It also excludes P. Lorenzen's generaliza- tion of multiplicative ideal theory (see, for example, ยง5 of W. Krull's Ideal- theorie, Ergebnisse der Math., Berlin, 1935) to any commutative semigroup S with cancellation, in which S (or its quotient group) is endowed with a family of subsets called r-ideals, satisfying certain conditions analogous to those for closed sets in topology. The book aims at being largely self-contained, but it is assumed that the reader has some familiarity with sets, mappings, groups, and lattices. The material on these topics in an introductory text such as Birkhoff and MacLane, A Survey of Modern Algebra (New York, Revised Edition, 1953) should suffice. Only in Chapter 5 will more preliminary knowledge be required, and even there the classical definitions and theorems on the matrix representations of algebras and groups are summarized. We have included a number of exercises at the end of each section. These are intended to illuminate and supplement the text, and to call attention to papers not cited in the text. They can all be solved by applying the methods and results of the text, and often more simply than in the paper cited. Each volume has a separate bibliography listing those papers referred to in that volume. No attempt has been made to list those papers on semi- groups to which no reference has been made in the text or exercises. The combined bibliography contains about half of the papers which have appeared in the (strictly) algebraic theory of semigroups. (The bibliography in Lyapin's book appears to be complete.) Whenever possible, the reference to the review of each paper in the Mathematical Reviews has been given, (MR x, y) denoting page y of volume x. English translations of Russian titles are those given in the Mathematical Reviews. The material in Volume 1 (more or less) was presented in a second-year graduate course at Tulane University during the academic year 1958-1959, and this volume has benefited greatly from the students' criticisms. The authors would also like to express their gratitude to Professors A. D. Wallace, D. D. Miller, and P. F. Conrad for many useful suggestions and, above all, to Dr. W. D. Munn for his very valuable criticisms, especially of Chapter 5,

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