A Survey of Binary Systems

The area of mathematics surveyed by the present report might be described as a little too broad to be treated in an encyclopedic manner in a reasonable space and yet not sufficiently developed in all its parts to warrant several distinct reports. Group theory and lattice theory are both regarded as outside the scope of the report. Subject to this restriction, the area surveyed is the algebraic theory of systems with a single binary operation. The operation is understood to be single- valued except in the case of multigroups and allied systems. Since quasigroups, for example, can be studied either in terms of a single binary operation or in terms of three, a little of the theory of general algebras enters quite naturally. A diagram indicating the interrelations of the various systemswill be found at the end of Chapter li.

The more highly developed parts of the subject have been treated in detail. Here the emphasis is on theorems and proofs; the work of several mathematicians has been fused into a single whole with only a mild attempt to indicate their several contributions. This is the case, for example, with the main body of the theory of loops and with the theory of ideals in semigroups. At other points, significant theorems have been stated without proof but with precise references to the literature. The author is weil aware that many special topics, for example in the theory of semigroups and quasigroups, have not been specifically mentioned in the text. The bibliography should partly make up for such omissions. I wish to acknowledge with thanks research grants from the J ohn Sirnon Guggenheim Foundation for the year 1946–47 and from the University of Wisconsin Research Committee for the fall of 1946 and the summers of 1952, 1954 and 1955- grants which were used partly to expand the subject matter and partly to delineate it.