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**A model theoretic oriented approach to partial algebras. Introduction to theory and application of partial algebras. Part I.**
*(English)*
Zbl 0598.08004

Mathematical Research, Vol. 32. Berlin: Akademie-Verlag. 319 p. M 40.00 (1986).

The book under review is the first part of two about partial algebras. The second part [Structural induction on partial algebras. Mathematical Research 18 (1984; Zbl 0553.08002)] has been written by H. Reichel.

As the author writes in his Preface: ”Despite a possible growing interest in features of partial algebras only few people have investigated or really used these structures so far, since most possible users had great reservations against these structures whose treatment seemed to be so much more complicated than the one of total algebras and for which they did not see much of a theory around. As a matter of fact almost every concept from the theory of total algebras often splits into more than three corresponding concepts for partial algebras, and for a long time it seemed hard to distinguish some rules behind it - but this has changed.”

The idea of a book on universal algebras which also treats partial algebras as fundamental structures had already been in the air just at the end of the sixties. During the last decade there has been developed a theory of partial algebras which may really be a starting point for the applications of these structures. This theory is based on category and model theoretical methods and provides some insight into the reasons for the fact that there really is such a wealth of concepts around for partial algebras. The fact that the theory of relational systems can totally be embedded into the theory of partial algebras and the fact that out of the theory of partial algebras one may get a better understanding of many-stored or heterogeneous algebras which are the structures mostly used in computer science - are only two of the reasons for this richness. Thus the reviewer agrees with the author in thinking ”to be worth-while to write a book like this one which introduces into the theory of partial algebras and shows up their special features in comparison to total algebras” and considers that the author succeeded quite well.

The book is an excellent introduction to this beautiful and difficult subject. It is relatively self-contained and collects material previously scattered in research literature. The book has an excellent bibliography. Each chapter is preceded by a brief survey of the contents in its sections, which enable readers to track down the references easily. Also, the list of symbols and the index of notions are very handy. These features and the selection and presentation of the material make this book a valuable tool.

Chapter I (Basic definitions and properties) presents a great part of the universal algebra properties of partial algebras. Starting with basic definitions and concepts the theory of partial algebras is developed up to their equational theory. Based on the investigations of homogeneous partial algebras heterogeneous partial algebras are included in the considerations.

Chapter II (A basis for two-valued model theory for partial algebras) is dedicated to the fundaments of the theory of universal Horn formulas based to a large extent on category theoretical methods.

The rôle of equations as most important axioms in the total case is taken over by elementary implications in the case of partial algebras. Category theoretical methods help to provide a meta-theorem which yields Birkhoff type theorems for a wealth of special kinds of elementary implications. This meta-theorem is the core of Chapter III (Birkhoff type results for elementary implications).

In the last chapter (Special topics) the author treats or extends some fundamental concepts which he either touched so far not at all or only very briefly: algebraic operations, semilattices of (weak) completions, independence, free partial algebras, primitive classes, dependence of operations on arguments, congruence relations, subdirect representations, etc.

The book ends with an Appendix collecting some concepts and results from set theory, theory of partial orders, lattices, closure systems and Galois correspondences, which are often used throughout the book.

As the author writes in his Preface: ”Despite a possible growing interest in features of partial algebras only few people have investigated or really used these structures so far, since most possible users had great reservations against these structures whose treatment seemed to be so much more complicated than the one of total algebras and for which they did not see much of a theory around. As a matter of fact almost every concept from the theory of total algebras often splits into more than three corresponding concepts for partial algebras, and for a long time it seemed hard to distinguish some rules behind it - but this has changed.”

The idea of a book on universal algebras which also treats partial algebras as fundamental structures had already been in the air just at the end of the sixties. During the last decade there has been developed a theory of partial algebras which may really be a starting point for the applications of these structures. This theory is based on category and model theoretical methods and provides some insight into the reasons for the fact that there really is such a wealth of concepts around for partial algebras. The fact that the theory of relational systems can totally be embedded into the theory of partial algebras and the fact that out of the theory of partial algebras one may get a better understanding of many-stored or heterogeneous algebras which are the structures mostly used in computer science - are only two of the reasons for this richness. Thus the reviewer agrees with the author in thinking ”to be worth-while to write a book like this one which introduces into the theory of partial algebras and shows up their special features in comparison to total algebras” and considers that the author succeeded quite well.

The book is an excellent introduction to this beautiful and difficult subject. It is relatively self-contained and collects material previously scattered in research literature. The book has an excellent bibliography. Each chapter is preceded by a brief survey of the contents in its sections, which enable readers to track down the references easily. Also, the list of symbols and the index of notions are very handy. These features and the selection and presentation of the material make this book a valuable tool.

Chapter I (Basic definitions and properties) presents a great part of the universal algebra properties of partial algebras. Starting with basic definitions and concepts the theory of partial algebras is developed up to their equational theory. Based on the investigations of homogeneous partial algebras heterogeneous partial algebras are included in the considerations.

Chapter II (A basis for two-valued model theory for partial algebras) is dedicated to the fundaments of the theory of universal Horn formulas based to a large extent on category theoretical methods.

The rôle of equations as most important axioms in the total case is taken over by elementary implications in the case of partial algebras. Category theoretical methods help to provide a meta-theorem which yields Birkhoff type theorems for a wealth of special kinds of elementary implications. This meta-theorem is the core of Chapter III (Birkhoff type results for elementary implications).

In the last chapter (Special topics) the author treats or extends some fundamental concepts which he either touched so far not at all or only very briefly: algebraic operations, semilattices of (weak) completions, independence, free partial algebras, primitive classes, dependence of operations on arguments, congruence relations, subdirect representations, etc.

The book ends with an Appendix collecting some concepts and results from set theory, theory of partial orders, lattices, closure systems and Galois correspondences, which are often used throughout the book.

Reviewer: C.Hatvany

### MSC:

08A55 | Partial algebras |

08-02 | Research exposition (monographs, survey articles) pertaining to general algebraic systems |

08C10 | Axiomatic model classes |

03C60 | Model-theoretic algebra |

68Q65 | Abstract data types; algebraic specification |

08B05 | Equational logic, Mal’tsev conditions |

03C05 | Equational classes, universal algebra in model theory |