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Preface
Universal algebra has enjoyed a particularly explosive growth in the last twenty years, and
a student entering the subject now will find a bewildering amount of material to digest.
This text is not intended to be encyclopedic; rather, a few themes central to universal
algebra have been developed sufficiently to bring the reader to the brink of current research.
The choice of topics most certainly reflects the authors’ interests.
Chapter I contains a brief but substantial introduction to lattices, and to the close connection between complete lattices and closure operators. In particular, everything necessary
for the subsequent study of congruence lattices is included.
Chapter II develops the most general and fundamental notions of universal algebra—
these include the results that apply to all types of algebras, such as the homomorphism and
isomorphism theorems. Free algebras are discussed in great detail—we use them to derive
the existence of simple algebras, the rules of equational logic, and the important Mal’cev
conditions. We introduce the notion of classifying a variety by properties of (the lattices of)
congruences on members of the variety. Also, the center of an algebra is defined and used to
characterize modules (up to polynomial equivalence).
In Chapter III we show how neatly two famous results—the refutation of Euler’s conjecture on orthogonal Latin squares and Kleene’s characterization of languages accepted by
finite automata—can be presented using universal algebra. We predict that such “applied
universal algebra” will become much more prominent.
Chapter IV starts with a careful development of Boolean algebras, including Stone duality, which is subsequently used in our study of Boolean sheaf representations; however,
the cumbersome formulation of general sheaf theory has been replaced by the considerably
simpler definition of a Boolean product. First we look at Boolean powers, a beautiful tool
for transferring results about Boolean algebras to other varieties as well as for providing a
structure theory for certain varieties. The highlight of the chapter is the study of discriminator varieties. These varieties have played a remarkable role in the study of spectra, model
companions, decidability, and Boolean product representations. Probably no other class of
varieties is so well-behaved yet so fascinating.
The final chapter gives the reader a leisurely introduction to some basic concepts, tools,
and results of model theory. In particular, we use the ultraproduct construction to derive the
compactness theorem and to prove fundamental preservation theorems. Principal congruence
ix.
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