We give here a brief preview of stability theory, as it underpins stable domination. We also introduce some of the model-theoretic notation used later. Familiarity with the basic notions of logic (languages, formulas, structures, theories, types, compactness) is assumed, but we explain the model theoretic notions beginning with saturation, algebraic closure, imaginaries. We have in mind a reader who is familiar with o-minimality or some model theory of valued fields, but has not worked with notions from stability. Sources include Shelah's Classification Theory as well as books by Baldwin, Buechler, Pillay and Poizat. There is also a broader introduction by Hart intended partly for non-model theorists, and an introduction to stability theory intended for a wider audience in. Most of the stability theoretic results below should be attributed to Shelah. Our treatment will mostly follow Pillay.

Stability theory is a large body of abstract model theory developed in the 1970s and 1980s by Shelah and others, but having its roots in Morley's 1965 Categoricity Theorem: if a complete theory in a countable language is categorical in some uncountable power, then it is categorical in all uncountable powers. Shelah formulated a radical generalization of Morley's theorem, weakening the categoricity assumption from one isomorphism type to any number less than the set-theoretic maximum. The conclusion is that all models of the theory, in any power, are classifiable by a small tree of numerical dimensions.

This appendix contains the complete Common Lisp implementation of the calendar functions described in the text. The functions in the text were automatically typeset from the definitions in this appendix. These functions are available over the World Wide Web at

http://www.cambridge.org/us/9780521702386

Please bear in mind the limits of the License and that the copyright on this book includes the code. Also please keep in mind that if the result of any calculation is critical, it should be verified by independent means.

For licensing information about nonpersonal and other uses, contact the authors. The code is distributed in the hope that it may be useful but without any warranty as to the accuracy of its output and with liability limited to return of the price of this book, which restrictions are set forth on page xxviii.

Lisp Preliminaries

For readers unfamiliar with Lisp, this section provides the bare necessities. A complete description can be found in [1].

All functions in Lisp are written in prefix notation. If f is a defined function, then applies f to the n + 1 arguments e0, e1, e2, … en.

The following description of the presentation of the annual calendar in China is taken from Peter (Pierre) Hoang (A Notice of the Chinese Calendar and a Concordance with the European Calendar, 2nd ed., Catholic Mission Press, Shanghai, 1904):

Much late 19th and early 20th century work in logic was in a 2nd order framework; infinitary logics in the modern sense were foreshadowed by Schroeder and Pierce before being formalized in modern terms in Poland during the late 20's. First order logicwas only singled out as the ‘natural’ language to formalize mathematics as such authors as Tarski, Robinson, and Malcev developed the fundamental tools and applied model theory in the study of algebra. Serious work extending the model theory of the 50's to various infinitary logics blossomed during the 1960's and 70's with substantial work on logics such as Lω1, ω and Lω1, ω(Q). At the same time Shelah's work on stable theories completed the switch in focus in first order model theory from study of the logic to the study of complete first order theories As Shelah in [44, 46] sought to bring this same classification theory standpoint to infinitary logic, he introduced a total switch to a semantic standpoint. Instead of studying theories in a logic, one studies the class of models defined by a theory.

Introduction. Several applications of model-theoretic methods in the theory of cohomology have appeared recently, most probably influenced by ideas of Macintyre. In his programmatic paper [10], he shows that, after expanding the language of rings by certain sorts and predicates for cohomology, the axioms of Weil cohomology theories are first order and one can form new cohomology theories as ultraproducts of already existing ones.

The present author has remarked that, although the cohomologies with torsion coefficients do not satisfy the axioms for a Weil cohomology theory individually, they do so “on average”, and one can obtain cohomologies with coefficients in pseudofinite fields of characteristic zero by taking ultraproducts [13]. He shows that this “pseudofinite cohomology” is at least as good as the l-adic theory when dealing with issues around the Weil conjectures. In parallel, Brunjes and Serpe developed the theory of nonstandard sheaves systematically, and they even show that the pseudofinite cohomology is better behaved than the l-dic one, the former being a derived functor cohomology [1].

The purpose of this short note is to clarify which aspects and invariants of the theory of (étale) constructible sheaves and cohomologies are definable in the language of rings. It should serve as a bridge between the algebraic-geometric and model-theoretic language and should encourage model-theorists to use the sophisticated techniques already developed by geometers. We show that, in case one needs to consider an invariant defined in terms of constructible sheaves over a finite or a pseudofinite ground field, there is a good chance that it is definable.

The International Organization for Standardization (ISO) calendar, popular in Sweden and other European countries, specifies a date by giving the ordinal day in the week and the “calendar week” in a Gregorian year. The ISO standard [1, sec. 2.2.10] defines the calendar week number as the

This does not define a new calendar per se, but rather a representation of dates on the Gregorian calendar; still, it is convenient for us to treat it as a separate calendar because the representation depends on weeks and the day of the week.

It follows from the ISO standard that an ISO year begins with the Monday between December 29 and January 4 and ends with a Sunday between December 28 and January 3. Accordingly, a year on the ISO calendar consists of 52 or 53 whole weeks, making the year either 364 or 371 days long. The epoch is the same as the Gregorian calendar, namely R.D. 1, because January 1, 1 (Gregorian) was a Monday.

I intend in this article to introduce nonspecialists to the fact that there are deep results in contemporary universal algebra. The first four sections give the context in which universal algebra has something to say, and describe some of the basic results upon which much of the work in the field is built. Section 5 covers the highlights of tame congruence theory, a sophisticated point of view from which to analyze locally finite algebras. Section 6 describes some of the field's “big” results and open problems concerning finite algebras, notably the undecidability of certain finite axiomatizability problems and related problems, and the so-called “RS problem,” currently the most important open problem in the field.

This article presents a personal view of current universal algebra, one which is limited both by my ignorance of large parts of the field as well as the likely interests of the intended audience and the need to keep the article focused. For example, I do not mention natural duality theory, one of the most vigorous subdisciplines of the field, nor algebraic logic or work that is motivated by and serves computer science. Therefore the views expressed in this article should not be taken to be a comprehensive statement of “what universal algebra is.”

The text is frequently imprecise and proofs, when present, are merely sketched. Credit is not always given where due. My aim is to give the reader an impression of the field. Resources for further reading are provided in the final section.