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Book review

Author(s) Adamek, J.
Trnkova, V.
Title Automata and Algebras in Categories
Publisher Kluwer Academic Publishers
Year of publication 1990
Reviewed by Ilie Parpucea

The original notion of a sequential automaton has been generalized in a number of directions. The motivating directions for the present volume are two: (i) linear sequential automata, arising from the theory of dynamical systems, and (ii) tree automata, the basic structure of which is an arbitrary algebra (whereas the structure of a sequential automaton is a unary algebra).

The authors discuss about the existence and construction of free F-algebras which play the role that the monoid of words does for sequential automata, the existence of minimal realizations for all behaviors, and their construction and universality, and the languages recognizable by finite deterministic and nondeterministic automata. The obtained results have diverse degrees of generality: in the "constructive" and "finitary" existence of free algebras the categories and functors are quite general, in the existence and universalty of minimal realizations are assumed restrictive additional hypotheses, and in the description of the languages recognized by finite automata using rational operations is studied only in the categories of sets.

The book is maked self-contained. All concepts of the theory of automata the readers use can be found in the first two chapters. The reader is expected to be familiar with the fundamentals of category theory.

All those interested in the theory of automata are advised to purchase this volume. This is a valuable book which most teachers of theory of automata would like to have within reach.

The authors of book are well-known researchers in the field of theory of automata.