Combinatorial Functors.by J. N. Crossley; Anil Nerode

Combinatorial Functors. by J. N. Crossley; Anil NerodeReview by: Carl E. BredlauThe Journal of Symbolic Logic, Vol. 42, No. 4 (Dec., 1977), pp. 586-587Published by: Association for Symbolic LogicStable URL: http://www.jstor.org/stable/2271884 .

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586 REVIEWS

syntax, whence Tarski's theorem follows immediately. A diagonal argument shows that 'theorem of T' is representable in no consistent T extending Q; one short additional argument shows that no such T can be complete and recursively axiomatized (the quick proof of Church's theorem appears here). A curious feature of this approach is that no undecidable sentence is produced until one gets to the second incompleteness theorem; w-consistency and Rosser's construction are circumvented; some of the lemmas have more of the flavor of a reductio than they do in the classical treatment. The second incompleteness theorem is derived as a corollary of the remarkable theorem of Lob; this is, I believe, the first text to include Lob's theorem; regrettably, the beautiful informal argument which underlies this result is omitted. One of the derivability lemmas is here quoted without proof-I should mention that this and the "proofs by Church's thesis" alluded to earlier are the only deviations in the book from full rigor.

The remainder of the book treats a less unified set of topics, loosely grouped around the notions of expressibility and decidability. Compactness and Skolem-L6wenheim are used to produce countable non-standard models of arithmetic; the authors prove that these all have the same reduct to the language { < }. By contrast, second-order arithmetic is shown to be categorical; there is here a good discussion of second-order logic. Then some results in hierarchy theory: 'true in arithmetic and of length ' n' is shown to be arithmetical for each n; the unit set of the truth set is proved arithmetical; the predicate 'arithmetical' is shown not to be arithmetical (Addison)-this requires a discussion of arithmetical forcing(!). The authors give what seems a rather more difficult proof of the Craig interpolation lemma than is needed; this in turn is used to show Robinson consistency and Beth definability theorems. The results on decidability presented here are as follows: decidability of arithmetic without multiplication; decidability of monadic logic; undecidability of dyadic logic, of logic with even one dyadic predicate. This part of the book is beautifully clear; I would, however, recommend it more as a reference than as the basis for a course; so many unrelated techniques are introduced and applied only once that the student may have trouble assimilating them all on a first exposure. The other parts of the book (possibly with the omission of the chapter on abaci), together with a few chapters from this section, should add up to a good semester's work.

There are a number of typographical errors, all fairly easy to catch-an up-to-date list is available from Boolos. Here are some non-typographical corrections. The definition presented of 'recursive' actually defines the partial recursive functions; conformity with standard usage does not begin until page 98, where it should be signaled as a change in terminology. The fact that more than one parameter is allowed in definition by primitive recursion is buried in an easily overlooked aside. Although the compactness theorem is correct as stated, only countable compactness is proved; accordingly, one might insert the word 'countable' in many places in Chapter 12. The theorem which, according to the first lines of page 162, was proved earlier was not proved; it is, however, an obvious consequence of the normal form theorem. STEPHEN LEEDS

J. N. CROSSLEY and ANIL NERODE. Combinatorial functors. Ergebnisse der Mathematik und ihrer Grenzgebiete, vol. 81. Springer-Verlag, New York, Heidelberg, and Berlin, 1974, VIII + 146 pp.

A very short review would be that Crossley and Nerode's Combinatorialfunctors is to category theory as Dekker and Myhill's Recursive equivalence types (XXV 356) is to set theory. The motivation behind the book comes from the observation that Dekker's recursive equivalence types, Crossley's constructive order types, and Dekker's a-spaces over a finite field have as their underlying structure an infinite model of an HO-categorical theory. These models are the natural numbers with equality, the rational numbers with the less-than relation, and an N,-dimensional vector space over a finite field, respec- tively. Moreover, each model has effectively computable, structure-preserving maps on subsets; in particular, the last model permits extensions to the space spanned by the

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REVIEWS 587

subset. "This led us to consider the notion of the algebraic closure of a set in a model.... With this in mind it seemed natural to ask about the recursive isomorphism types of algebraically closed subsystems of a fully effective, but otherwise arbitrary, N0t- categorical model."

The book, in ten parts, generalizes theorems about combinatorial operators and functions, recursive equivalence types and isols, solutions of equations of combinatorial functions, and extensions of relations via frames to suitably defined categories. About a third of the book consists of developing the necessary definitions and theorems for the category and model of an HO-categorical theory. The remainder consists of proving the generalizations of theorems about recursive equivalence types. Most of the tersely written proofs are similar in construction to the proofs of the original theorems.

The same material is outlined in the authors' paper in Logic colloquium, Symposium on logic held at Boston, 1972-73 (Springer-Verlag, 1975), pp. 1-2 1. CARL E. BREDLAU

RAYMOND BALBES and PHILIP DWINGER. Distributive lattices. University of Missouri Press, Columbia 1974, xiii + 294 pp.

This monograph provides an extensive treatment of the fundamentals of the theory of distributive lattices, together with applications of the theory to special classes of lattices. The emphasis of the book is on representation and structure theory. The detailed contents are as follows. Chapter I gives appropriate background in universal algebra and category theory, Chapter II the elementary algebraic theory of lattices. The next five chapters are an exposition of the basic structure and representation theory for distributive lattices and Boolean algebras. Chapter V discusses some universal construc- tions involving extensions of maps, such as free algebras, free Boolean extensions, injectives, and projectives. Chapter VI deals with lattices which are subdirect products of chains of a given finite length. The problem of a complete characterization of these lattices is still open, but the authors give their own characterization of subdirect products of the three-element chain. Chapter VII treats of coproducts and colimits. Chapters VIII to XI discuss lattices related to some non-classical logics. The topic of Chapter VIII is pseudocomplemented distributive lattices, including Glivenko's theorem, and Stone algebras. Chapter IX is a somewhat brief discussion of Heyting algebras; representation as lattices of open sets and the structure of the one-generated free algebra are included. Chapters X and XI give a fairly extensive exposition of Post algebras, Lukasiewicz algebras, and De Morgan algebras. Chapter XII is on complete and a-complete distributive lattices; it includes theorems on normal completions and on representations preserving certain infinite meets and joins.

The book differs from similar works on lattice theory by its emphasis on category theory as a unifying language. The algebraic, topological, and category-theoretic methods do not, however, always mesh as smoothly as they might. A more unified presentation would have been possible had the authors adopted Priestley's representa- tion of distributive lattices by ordered topological spaces (The bulletin of the London Mathematical Society, vol. 2 (1970), pp. 186-190) rather than the original topological representation theory of Stone. Priestley's version is preferable not only by reason of its simplicity but also because many topics such as ideals, coproducts, Boolean extensions, and pseudocomplementation find their natural topological setting in this theory. A result which can be stated most readily in terms of category theory is the well-known duality between the category of finite bounded distributive lattices and finite partially ordered sets. This duality is an immediate consequence of the Priestley theory. However, this result does not appear anywhere in the present book, presumably because of the emphasis on Stone's original representation.

The authors make a common historical error in crediting Nishimura (XXXII 396) with the first description of the free Heyting algebra with one generator. This result is in fact due to Rieger (XVII 146).

In spite of the above criticism the authors are successful in their stated aim of

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