Postscript

Summary

In this postscript, we intend to explain somewhat the position our book has in the literature on algebra and category theory, and we want to mention some of the important topics that we decided not to deal with in our book.

One-sorted algebraic theories provide a very convenient formalization, based on the concept of finite product, of the classical concept of “the collection of all algebraic operations” present in a given kind of algebras, for example, in groups or boolean algebras. These theories lead to concrete categories A of algebras, that is, to categories equipped with a faithful functor U: A → Set. They can also be used to find an algebraic information present in a given concrete category A: we can form the algebraic theory whose n-ary operations are precisely the natural transformations Uⁿ → U. In the case of groups (and in any one-sorted algebraic category), these “implicit” operations are explicit; that is, they correspond to operations of the theory of groups. But on finite algebras (e.g., finite semigroups), there exist implicit operations that are not explicit, and they are important in the theory of automata (see Almeida, 1994). The passages from one-sorted algebraic theories to one-sorted algebraic categories and back form a duality that is a biequivalence in general. And, as we will see in Appendix C, this passage is an equivalence if we restrict one-sorted algebraic categories to uniquely transportable ones.