The book aims at introducing the reader to the theory of function algebras, often called clones, and provides the state of the art for part of this theory. A clone on a set \(X\) is a set of operations on \(X\) of finite arity which contains all projection functions and which is closed under superpositions. Equivalently, a clone can be defined as the set of term operations of an algebra with universe \(X\). The author considers function algebras only for the case where \(X\) is finite. So far, the only extensive works on this subject were the German book [R. Pöschel and L. A. Kalužnin, Funktionen- und Relationenalgebren. Ein Kapitel der diskreten Mathematik. Berlin: VEB Deutscher Verlag der Wissenschaften (1979; Zbl 0418.03044); Lizenzausgabe: Basel, Stuttgart: Birkhäuser Verlag (1979; Zbl 0421.03049)], and the monograph [Á. Szendrei, Clones in universal algebra. Montréal (Québec), Canada: Les Presses de l’Université de Montréal. (1986; Zbl 0603.08004)], which date back to 1979 and 1986, respectively.
The first part of the present book is a 65-page introduction to universal algebra. It gives the definition of a universal algebra, puts most commonly studied algebras, such as groups and vector spaces, into this context, and introduces lattices, closure operators, and varieties. The second and major part, which rarely refers to notions of the first part, has around 550 pages and deals with function algebras (clones). They are understood as subalgebras of the full function algebra \(P_X\) over \(X\), which has all finitary operations on \(X\) as its universe, and which is equipped with finitely many operations that guarantee that the subalgebras of \(P_X\) are exactly the clones on \(X\). This approach is reflected in the selection of the topics for the rest of the book: The author presents numerous results which list specific clones and their properties, while not putting much emphasis on the global structure of the lattice of clones. For example, she considers the question which automorphisms a single clone (as a function algebra) has, rather than the question of finding the automorphisms of the whole clone lattice.
The book contains: A thorough description of Post’s lattice, a traditional proof of Rosenberg’s characterization of the maximal clones, other completeness criteria for algebras, a list of many submaximal clones (i.e., clones which are covered by a maximal one), as well as investigations of clones of linear functions and of clones of functions which take only two fixed values of \(X\). Moreover, it deals with congruences and automorphisms of clones as function algebras, with the relation degree and order of certain clones, the question how many subclones submaximal clones contain, and partial clones. There is only a very short section on minimal clones. Generally, the author concentrates on the upper part of the clone lattice. Applications of clones to universal algebra and theoretical computer science (e.g, the constraint satisfaction problem) are only briefly mentioned.
The book does not contain: Monoidal intervals, lattice-theoretic aspects (such as sublattices of the clone lattice), symmetric clones, clones on infinite sets.