An Overview of Modern Universal Algebra

This article, aimed specifically at young mathematical logicians, gives a gentle introduction to some of the central achievements and problems of universal algebra during the last 25 years. 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 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. References are often omitted. My aim is to give the reader an impression of the field. Resources for further reading are provided in the final section. 1. Algebras and equational classes The fundamental objects of study in universal algebra are algebraic structures, or algebras for short. These are structures in the traditional sense of model theory but with no relations. The algebras in this article can be assumed with no loss of generality to have a countable language, and often the language will be finite. Notationally I will use A to denote an algebra with universe A, and will write A = 〈A; f0, f1, . . .〉. Unlike model theory, which is primarily interested in the structure 2000 Mathematics Subject Classification. Primary 08-02; Secondary 03C05.