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Clones in universal algebra. (English) Zbl 0603.08004
Séminaire de Mathématiques Supérieures. Séminaire Scientifique OTAN (NATO Advaned Study Institute), 99. Département de Mathématiques et de Statistique, Université de Montréal. Montréal (Québec), Canada: Les Presses de l’Université de Montréal. 166 p. $ 19.00 (1986).
This is a well-written Seminar Note. Everything is properly explained and easy to understand. It is useful both for beginners and for those who know the topic.
From the introduction: ”... most properties of algebras depend on their term operations rather than on the choice of their basic operations. During the last fifteen years or so... the application of new algebraic methods produced a rapid development, and by now the theory of clones has become an integral part of universal algebra. The aim of these lecture notes is to introduce the reader to some results showing how clones can contribute to the understanding of the structure of algebras, and not less importantly, to present several techniques in clone theory.”
The author considers the topics she is most interested in. The contents shows that this is not a narrow choice, at all: The book consists of the following chapters: 1. Algebras and clones; 2. Affine and semi-affine algebras; 3. Unary term operations in algebras; 4. Quasi-primal and para- primal algebras; 5. Homogeneous algebras; 6. Functionally complete algebras.
The book, generally, does not contain new results. The only real exception is ”Idempotent non-quasi-primal algebras” in Chapter 4, which ”... prepares the study of para-primal algebras as well as the description of homogeneous algebras...” (Unfortunately the letter \({\mathfrak A}\) is missing from Theorem 4.3.) Quite a few proofs are simpler and theorems stronger than the original ones. She uses new methods, too. It is a pity that there is no index to the book.
Reviewer: E.Fried

MSC:
08A40 Operations and polynomials in algebraic structures, primal algebras
08-02 Research exposition (monographs, survey articles) pertaining to general algebraic systems