Ko Ker-I. Complexity theory of real functions. Progress in theoretical computeer science. Birkhäuser, Boston, Basel, and Berlin, 1991, viii + 309 pp.

ANDREW B. CARSON. Model completions, ring representations and the topology of the Pierce sheaf. Pitman research notes in mathematics, no. 209. Longman Scientific and Technical, Harlow, Essex, and John Wiley & Sons, New York, 1989, vi + 107 pp. For certain classes of rings there are representation theorems of the type that any ring R in the class is isomorphic to a ring of continuous functions on a space X associated with the ring (its spectrum), with values in a particular ring F of a given type. Such are, for example, the Stone representation theorem for Boolean algebras and the Gelfand representation theorem for C*-algebras. In general, a single ring F does not suffice and it is necessary to resort to sheaves of rings of a certain type on X, with the ring R being represented by the continuous global sections of one such sheaf. Among these are, for example, the Pierce representation theorem for von Neumann regular rings or the Keimel representation theorem for /-rings. Despite the success of sheaves in the representation theory of rings, there is a certain preference among mathematicians working in this area for representations resembling the simple type, that is, involving continuous functions on X with a fixed codomain rather than those involving global sections of a sheaf on X. The present monograph can be said to belong to such a nostalgic movement. It aims at generalizing a previous theorem of the author's stating that, if R is a von Neumann regular algebra over a field L, if R is algebraic over L, and if the Boolean algebra of idempotents of R is either countable or complete, then R can be represented as the ring of certain specified continuous functions on dense open subsets of the Boolean spectrum of R with values in the algebraic closure of L. The generalization in question consists in removing the condition that R be algebraic over L as well as the conditions on B, and it is the principal goal of the monograph. For the logician, the main interest of sheaf representations lies in its model-theoretic applications, exemplified by the theorems of the author and of Lipshitz and Saracino for von Neumann regular rings, and of van den Dries for /-rings. In both of these cases, a model completion is found by transfer from the theory of the stalks to the theory of the global sections. The method itself is abstracted in the celebrated theorem of Macintyre's concerning model completeness, although other model-theoretic properties (amalgamation, prime model extensions) can be handled in a similar manner. (See XLVIII 496.) In the setting of the monograph being reviewed, no such transfer theorems are directly applicable. The author does, however, prove a general result on the existence of model completions (in a suitable language) for a wide class of theories of rings that are represented, not as rings of all global sections of a sheaf on X of rings of a certain type, but as rings of specified continuous functions on X having (on the other hand) values in a fixed ring F of a certain type. Not surprisingly, among the assumptions made about such theories of rings are both the validity of Macintyre's theorem and that of Comer's version of the Feferman-Vaught lemma—the latter relating truth in the (stalks of) the sheaf on a Boolean space to truth in the global sections of the sheaf. Applications include division rings and many theories of not necessarily semi-prime commutative rings. There are quite a few demands made on the reader of this monograph, going beyond a thorough acquaintance with the Pierce representation and with the theory of model completions. Although a claim of self-completeness is made by the author for tools employed in the process of achieving his main goal, it seems to the reviewer that a certain familiarity with torsion theories and with forcing techniques (meaning the language of a topos of sheaves for the category theorist) is desirable. Without those, the reading of this highly technical work could be tough going, particularly since general discussions and motivational remarks do not abound. The wide variety of methods employed by the author makes his work very interesting. On the other hand, the reviewer regrets to find that the methods of category theory, which can be and have been illuminating in the field at hand, are absent from the work. A final comment is that, although the main result is about theories of rings, the whole subject in general, and this book in particular, is more likely to appeal to logicians than to ring theorists.