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The clone of a topological space. (English) Zbl 0615.54013
Research and Exposition in Mathematics, 13. Berlin: Heldermann Verlag. V, 91 p. DM 40.00 (1986).
Motivated by the fact that many properties of a topological space are reflected in the behavior of functions of more than one variable defined on A, the author explores the graded algebra $$C(A)=C_ 0\cup C_ 1\cup C_ 2\cup..$$. where each $$C_ n$$ is the family of all maps $$A^ n\to A$$. This algebra C(A) is called the clone of A. (The definitions can be made for A an object in any category with finite direct powers, but the focus here is on the category of topological spaces and continuous functions. One general question posed is: Which clones are isomorphic to C(A) for some space A?)
This is a survey article that the author terms as exploratory with many of the results being interesting but not definitive. There is overlap with an earlier survey [W. Taylor, Can. J. Math. 29, 498-527 (1977; Zbl 0357.08004)]. An extensive bibliography of related works in topology, topological algebra, and universal algebra is included.
In the spirit of determining the topological space from its semigroup of selfmaps [K. D. Magill, jun., Semigroup Forum 11, 189-282 (1975; Zbl 0338.20088)] or its ring of continuous functions [L. Gillman and M. Jerison, Rings of continuous functions (1960; Zbl 0093.300)], the thrust is toward determining those properties of A that are first order clone properties, that is, those that can be expressed by a formula in the elements of C(A). Two kinds of clone-theoretic properties are emphasized: (1) satisfiability of a set of identities, and (2) universality of a set of terms.
The first of these can be interpreted in C(A), for A a topological space, as identities can be modeled on A by continuous operations. That is essentially answers to the question asked by A. D. Wallace [Bull. Am. Math. Soc. 61, 95-112 (1955; Zbl 0065.008)], ”What spaces admit what structures?” This work discusses the clone identities yielding various kinds of H-spaces, semigroups, and groups, as well as more complex algebraic structures.
The first result on the universality of a set of terms is: If a set of terms is universal on a space A, then there is a set of identities that cannot be satisfied by continuous operations on A. If A is either totally disconnected or both completely regular and arcwise connected, then the converse holds. Several topological properties are shown to be equivalent to the universality of a certain set of terms. Among these are (sometimes with restrictions on the spaces): connectedness, fixed point property, product indecomposability, and indecomposability.
Many topological properties are described by unrestricted first order formulas for clones. These properties include arcwise connectedness, compactness, and total disconnectedness. Further, the homology and cohomology groups of A and the ring of continuous functions on A are defined in terms of the clone C(A). The survey concludes with several open questions.
 54C40 Algebraic properties of function spaces in general topology 54C35 Function spaces in general topology 54C05 Continuous maps 55Q05 Homotopy groups, general; sets of homotopy classes 54-02 Research exposition (monographs, survey articles) pertaining to general topology 55N99 Homology and cohomology theories in algebraic topology 17A40 Ternary compositions 17A42 Other $$n$$-ary compositions $$(n \ge 3)$$ 16W80 Topological and ordered rings and modules