Topologically Inseparable Functions I: Finitary Case
نویسندگان
چکیده
Given a finite set A and a distinguished function f : A −→ A , we study the set of all functions g : A −→ A that are continuous for all topologies for which f is continuous. The main result is a characterization of the functions f such that this set is trivial, that is, contains only the constant functions and the iterates of f . Introduction The Problem. Let A be a set and f : A → A be a fixed function. Associated with this function in a very natural way, we have the (algebraic) semigroup S0(A,f) generated by f S0(A, f) = {f : n ∈ N}, where f 0 is the identity function and for n ∈ N , f = f ◦ f . For any given topology τ over A , we have another associated semigroup, S(τ) , the semigroup of all τ–continuous functions on A . Suppose now that our distinguished function f is τ–continuous. Then any function in S0(A, f) , as well as all constant functions will be τ–continuous, in some sense, very trivially so. Are there any other functions that are also τ–continuous? A simple cardinality argument shows that this is usually the case. What happens if we change to another topology τ ′ for which f is continuous? Then, the constant functions and the members of S0(A, f) will still be continuous but the other continuous functions will in general be different. A very natural question arises then: Date: July 24, 2002. 1991 Mathematics Subject Classification. Primary: 54C05, Secondary: 54H10, 08A40. Funding for the first and third authors has been provided by DIUCT, grant 95–3–03. Funding for the second author has been provided by FONDECYT grant 1020621. 1 Are there any non–trivial functions that are continuous for every topology for which f is continuous? In other words, are there functions, other than the obvious ones mentioned above, whose continuity is implied by the continuity of f ? For a very simple example, let f : {0, 1, 2} −→ {0, 1, 2} be the cycle (0 1 2) . Then there are only three iterations of f , namely, f 0 , f 1 and f 2 . Nevertheless, it is easy to check that no matter what the topology on {0, 1, 2} is, if f is continuous, then any function from {0, 1, 2} into {0, 1, 2} is continuous. The reason for this is that if f is continuous then the topology has to be either trivial or discrete. So there are non–trivial functions that are forced to be continuous if f is continuous. We define S(A,f) = {g : A → A : g is continuous for all topologies for which f is continuous} = ⋂ {S(τ) : f is τ–continuous}. S(A, f) is a semigroup that contains S0(A,f) . We call it the semigroup of functions topologically inseparable from f . We let S(A,f)∗ be the set of all non constant elements of S(A, f) , and since constant functions are always continuous, regardless of the function f , we will usually work in the context of this set. We also let S c 0 (A,f) = S0(A,f) ∪ {constant functions on A} . This is the first of two papers in which we determine necessary and sufficient conditions on f so that S c 0 (A,f) = S(A,f) . In this first one we restrict our attention to the case when A is a finite set. In the second paper (see [1]) we study the infinite case. The techniques used in the finite and the infinite cases are quite different, this is the main reason why we divided this study into two separate articles. In the second paper we also study the clone of all n–ary functions on A that are topologically inseparable from f and we extend our results to it. Motivations. Even though the problem is quite natural and stands on its own, it has universal–algebraic origins and motivations and is inspired by [2]. Given a topological algebra A = 〈A; fi〉i∈I , there are (at least) two clones naturally associated to it. One is the clone Clo(A) , of all terms and the other is the clone Clo(A, τ) , where τ is the underlying topology, of all n–ary τ–continuous functions on A . 2 We observe that since A is a topological algebra, any term defined function g : A → A , (i.e., g = σA for some n–ary term σ ), is continuous (in the appropriate product topology); the identity and the constant functions are also continuous. Again, all these are trivially so. So Clo(A) is isomorphically embedded in Clo(A, τ) . Are these two clones “the same”, that is, isomorphic? The example above may be interpreted in this context as the case of a mono–unary algebra 〈{0, 1, 2}; f〉 . It amounts to the fact that Clo1(A) is not the same as Clo1(A, τ) . The answer to the problem stated in these two papers may give us an idea of how to solve the following question. Given a class of algebras, is its clone of terms representable by the clone of continuous functions of a certain topological space? We know for instance that the clone of terms for Boolean algebras is represented by the clone of continuous functions on {0, 1} with the discrete topology. 1. Preliminary Results We first recall that the relation on A defined by x ∼ y iff there exist n,m ∈ N such that f(x) = f(y) is an equivalence relation. The equivalence classes are called connected components or orbits. The reader can easily figure out what the orbits may look. Some examples appear in the following diagram.
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