Clones in Topology and Algebra
نویسنده
چکیده
Clones of continuous maps of topological spaces and clones of homomorphisms of universal algebras are investigated and their initial segments compared. We show, for instance, that for every triple 2 ≤ n1 ≤ n2 ≤ n3 of integers there exist algebras A1 and A2 with two unary operations such that the initial k-segments of their clones of homomorphisms are equal exactly when k ≤ n1, isomorphic exactly when k ≤ n2 and elementarily equivalent exactly when k ≤ n3. 1. Concepts and Results 1.1. According to [5], a clone Clo(X) on a nonvoid set X is a collection of finitary operationsX −→ X with n ∈ ω (where ω denotes the set of all finite ordinals), containing all Cartesian projections p (n) i : X n −→ X with i ∈ n = {0, . . . , n− 1}, that is, maps p (n) i (x0, . . . , xn−1) = xi, and closed under all operations C n m withm,n ∈ ω (called compositions in [5]), defined for every m-tuple f0, . . . , fm−1 : X n −→ X of elements of Clo(X) and any g : X −→ X in Clo(X) with the result f : X −→ X given by f(x0, . . . , xn−1) = g(f0(x0, . . . , xn−1), . . . , fm−1(x0, . . . , xn−1)). It is often convenient to regard Clo(X) also as a category k whose objects are all finite powers X,X,X, . . . of X, the set of k-morphisms X −→ X consists of all n-ary operations of Clo(X) including all projections p (n) i , the set of k-morphisms X n −→ X consists of all maps f = f0×̇ · · · ×̇fm−1 where (f0, . . . , fm−1) is an m-tuple of n-ary operations in Clo(X) and f is defined by f(x0, . . . , xn−1) = (f0(x0, . . . , xn−1), . . . , fm−1(x0, . . . , xn−1)), Received March 7, 1997. 1980 Mathematics Subject Classification (1991 Revision). Primary 54C05, 08A10.
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