Notes on Filters: Topological vs. Electrical (File C:/vps tex/book/filters ideals.tex)

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1 Notes on Filters: Topological vs. Electrical (File C:/vps tex/book/filters ideals.tex) Virendra P. Sinha July 2005 Words pass from the common vocabulary to a technical one through a chain of intuitive analogies. Then, over the years, they change interpretation, through generalizations, and acquire defined meanings that have no apparent connections with the common interpretation. Filters and ideals of algebra and topology, the large and the small of category theory, open sets of topology, spectrum of physics and mathematics and engineering, are pointed examples. Notes on Topology: September, 1976; also see Book A, p.30 The first chapters of both [Thron] and [Bushaw] are very illuminating. While [Thron] begins with a very illuminating introductory chapter what is topology, [Bushaw] starts with a historical introduction. Both discuss how notions of topology have evolved. The works of Cantor, Frechet, Hausdroff and others are discussed. The term topology perhaps first appeared [Thron, p.17] in the title of the book Vorstudien Zur Topologie by Listing in Filters were introduced by Cartan in 1937 [Thron, p.78, Chap. 13]. A filter can be likened to an ideal. A subset of a ring, which is closed under addition and is also closed under multiplication by any element of the ring is called an ideal. Analogously, a filter is a subset of the power set P(X) of a set X, which is closed under intersection and under union with arbitrary elements of P(X). A filter can be looked upon as the dual of an ideal. Expand this point and add examples. Filters and Ultrafilters December 9, 1985 Encoded from the hand written collection Notes on Filters and Ultrafilters of I say on the cover page: Filters and Ideals: Elaborate on its double meaning. Notes written on : Illustrations for ultrafilters Consider a set I = {a, b}, with power set P(X) = {, {a}, {b}, {a, b}} Construct a filter, F on I. Recall that F is a filter if F P(X) such that

2 1. A F and A B = B F 2. A F, B F = A B F 3. F, I F Example 0.1 F = {I} is a filter. Example 0.2 Construct a set G recursively, testing for the three conditions for being an ultrafilter. Let us say {a} G. Now, since {a} {a, b}, {a, b} G. Further, {a} {a, b} = {a} G. Therefore G = {{a}, {a, b}} is a filter. Example 0.3 Likewise, G = {{b}, {a, b}} is a filter. Aside 1 Recall in this context that {{a}, {a, b}} is the ordered pair a, b, and likewise, {{b}, {a, b}} is the ordered pair b, a. (Weiner, Kuratowski) Example 0.4 Is G = {{a}, {b}} a filter? No, because {a} {b} =, which is not in G, and conds 2 and 3 are violated. Aside 2 For examples 3 and 4, look at the Venn diagrams. Could we infer in general that for a set of subsets to form a filter, its Venn diagram must consist of concentric circles (or concentric closed curves). Exercise 0.1 Examine the notion of isomorphisms for filters. Amongst the filters constructed above, are there isomorphic ones? Ultrafilters If I recall right, I had not yet located Hammer s work. It was still puzzling to me why such a topological entity should be called a filter or ultrafilter. There was still the feeling that there has to be a link between what we call a filter, and what it means for topologists following Cartan. See notes connected with Cartan s ideas in one of the books DSPDMS.. The idea of working out the details for topological filters was to see if any connections become visible en route. (Comments: July 2, 2005). A filter F is an ultrafilter if for any other filter G, F G implies that F = G. Example 0.5 For I = {a, b}, consider the filter F = {{a}, {a, b}}, and another filter G = {{b}, {a, b}}. No other filter, but F G. So F is an ultrafilter. (Is this argument correct?) Example 0.6 I = {a, b, c}, F = {{a}, {a, b}, {a, b, c}}, F 1 = {{a}, {a, c}{a, b, c}}, F 2 = {{a}, {a, b}, {a, c}, {a, b, c Is F a filter? F 1 and F 2 are filters. F F 2, but F = F 2. Rather F F 2. So F is not an ultrafilter. But F 2 is an ultrafilter. Filters: Davis Example 0.7 Let A 0 I, A 0 =. Then F = {A P(I) A 0 A} is a filter. Example 0.8 For any x, let A 0 = {x}. Then = {A P(I) A 0 a} is an ultrafilter. For, if F F 1, then A 0 F 1, and if F 1 is a filter then for any A 0, A 0 A, A F 1. But by definition, we also have A F. Thus, for any x F 1, x F, i.e., F 1 F. That is, F = F 1. Davis shows this as follows. If F F 1 and A F 1 F, then x / A (for otherwise A 0 = {x} A, and hence A F). Thus A {x} = F 1, so that F 1 can not be a filter. (Comments: I do not like this proof.)

3 Proposition 0.1 If F 0 is a filter on I, then there is an ultrafilter F on I such that F F 0. Davis comments that it can not be proved from the axioms of set theory without the axiom of choice. It is, he adds, technically speaking, a weak form of the so called axiom of choice. Definition 0.1 G P(X) is a filter basis on I if P1 / G P2 A, B G = A B G P3 G = Example 0.9 Consider I = {a, b, c}. Start with two appropriately chosen subsets of I as members of G. They must have common members, for otherwise the intersection is, which is not permitted by definition. So let us start with {a}, {a, b}. G = {{a}, {a, b}} meets the thre conditions; it is not empty, it does not contain the empty set, and, of its two members, the intersection is {a}, which is in G. Thus G is a filter basis. Aside 3 July 2, But why call such a thing a basis? I do not think I had then found the answer. In retrospect, these results might have some application in morphological filtering, connected as they are with complete lattices. Proposition 0.2 If G is a filter basis on I, then there is a filter F on I such that F G. PROOF: Let F = {A A I, A C for somec G}. Every C G is also in F. Thus G F. Further, if A F and B A, then for some C G, A C. Then B C. Therefore by definition B F. That is, for A F and A B, B F. Next, for A, B F, there are C 1, C 2 G such that C 1 A and C 2 B. Then C 1 C 2 G. Then, again by definition, A B F. Thus A, B F means A B F. Finally, if F then there is C G such that C. But then by definition of empty set, C =. That is, G. But this can not be by definition for G. So F. Proposition 0.3 If F is an ultrafilter on I, and A I, then A F, or I A F, but not both. The 83 October Proposal Re-Examined; September 22, 1985 X : Power set of X X: Set of particles P: Subset of X; P X P δ : {x x P and size of x δ P δ : {x x P and size of x > δ δ = (x), : X R +, a valuation (X, ) a lattice homomorphic to (R +, ) under Definition 0.2 A filter f δ is a morphism f δ : X X such that F1 f δ (P) = P δ F2 f δ (P R) = f δ (P) f δ (R) F3 f δ (P δ ) = P δ

4 F4 f δ (P δ ) = This definition is an attempt to capture the intuitive commonplace idea of a filter in a manner that may connect it with the topological filters. For such connections, also see the other notes. Now, are F1 to F4 independent? Consider F2. We have f δ (P R) F1 = (P R) δ = {x x P R, and size of x δ} = {x x P, and size of x δ} {x x R, and size of x δ} = P δ R δ Thus F1 implies F2. So we can delete F2. Now consider F3. f δ (P δ ) F1 = (P δ ) δ = {x x P δ, and size of x δ} = {x x P, and size of x δ} = P δ Alert!! check up the third step above. July 3, 2005 Thus F1 implies F3. Now consider F4. f δ (P δ ) = (P δ ) δ = {x x P δ, and size x δ = {x x P, and size x > δ and size x δ} = Aside 4 Back to the question: What is size? Think of it as a map from a set X with a partial order to R + such that for any x, y X, if x V X and (y) (x) then y V, and if x V and y V then (y) > (x). Comment So F1 also implies F4! The definition of a filter simply reduces to the following. Definition 0.3 A filter f δ is a morphism f δ : X X such that f δ (P) = P δ for every P X Corollary 0.4 A filter f δ is idempotent, i.e., f δ f δ = f δ. Aside 5 Can we characterize a morphism by its invariant subsets, just as we characterize a linear transformation by its invariant subspaces. The above analysis is full of holes. Clean up. (Comment ) Aside 6 For more on abstracting the idea of filters, starting with the commonplace notion, see Book 78 Book 2 p.237. Another way of looking at a filter is to think of it as a device that separates out members of a set into equivalence classes based on its symmetries

5 Notes written on : How did it (filter notion in math) all start? What is the significance in the study of mathematical structures in general? How does the notion of filters, as introduced in topology, get generalized to ordered structures, such as those of lattices and boolean algebras? The old question about electrical filters (and similar other filters, even sieves) I am inclined to believe can be given a topological interpretation; they seem to have a connection with the notion of topological filters. The choice of the same name in two independent context, is it just an accidental coincidence, or is there more to it? It is my feeling that concepts and theories in general, whether empirical or formal, carry within them and amongst them a structural unity owing to the fact that they are all the result of the activities of the human mind, a result of reflection and contemplation of the same era (Refer Whitehead, Harvard the Future). It seems reasonable to suppose that mental constructs, whether prompted by our perception of the external world or pure mental afterthought, must follow the same kind of inner logic, one on which the metalanguage in which the mind works out theories is based. It is equally reasonable to suppose as a consequence that a mathematician conceptualizes so often primarily in response to his perceptions of mathematical objects, whereas an engineer conceptualizes in response to his perception of the way his physical devices work. There is good ground for their intuition to have a great deal of commonality. It is not surprising then to find that once in a while the mathematician and the engineer or the physicist independently come to use the same word to denote apparently different conceptual objects, prompted by altogether independent circumstances but which on subsequent investigation turn out to be structurally analogous. The choice of the term spectrum is a case in point. Hilbert s choice of the term spectrum was, as recounted in [Steen], purely in a mathematical setting in the context of studying operators and transformations. Physicists and electrical engineers on the other hand thought of using the word in the context of representations of measurements by systems treated as operators. The two independent usages ultimately turned out to imply the same structural entity (apparently to Hilbert s surprise). Looking back, one can t help feeling that it could not have been otherwise because the attributes that mathematicians were wanting to capture can be seen to to be entirely the same as those under study by physicists and engineers; only the specific contexts were different. One could say that the abstract formal object was the same at the meta level, and only the models were different. Davis on Filters Again Coded July 2, 2005 Davis s comment that the theorem Every filter can be embedded in an ultrafilter is a weak form of the axiom of choice prompted me to look up (Moore-axioChoice) for more historical details. It is a good source. A good historical account of how the concept evolved is contained there (pp ). A summary of Moore s account is as follows. 1 Independent efforts to generalize the notion of convergence of sequences in a topological setting led to the introduction of the concepts of directed sets and filters. In 1922, E.H. Moore (Univ. of Chicago) and H.C. Smith (Univ. of Phillipines) introduced the notion of directed sets. Their rather obscure presentation was recast in a form more suitable for topology by Birkhoff in In Birkhoff s formulation, one started with a directed set {x a } a A, i.e., a one one function with a transitive relation on A such that 1 I think these notes I made by extracting comments from the original text, retaining some of the key statements, and introducing here and there my own comments, without making a clear demarcation. It is interesting that I do not now recall whether I understood it all.

6 for every a, b A, there was some c A such that a b and b c. Such a set {x a } a A is said to converge to x, x a x, if for every open set V containing x, there exists some b in A such that b c implies x c V. As one can see, this generalizes the classical notion of a limit point by means of a partial order. In this way the notion of denumerability is avoided. The second notion that of a filter that was an outcome of efforts to generalize the notion of convergence, was introduced by Henri Cartan in his proposal to the Paris Academy of Sciences in He defined a filter on a set E to be a nonempty family F of subsets of E such that A1 / F A2 a, b F then a b F A3 If a b E and a F then b F The central notion in his construction was that of an ultrafilter, roughly a filter that was maximal. He showed, using the well ordering theorem and transfinite induction, the fundamental result now known as the ultrafilter theorem: Every filter can be extended to an ultrafilter. He applied this result in the characterization of compactness on a Hausdorff space. To be specific, he showed that a Hausdorff space E is compact if and only if every ultrafilter on E has a limit point. He defined a limit point p to be limit point of F if F contains every neighbourhood of p. Apparently, Cartan was not aware of the connections with boolean algebra and of the fact that what he called a filter was in fact the dual of an ideal in a boolean algebra, and that the ultrafilter theorem is equivalent to the boolean prime ideal theorem. Task Set: Connect with 1. Fields s Nominalism 2. Hammer s extended topology 3. Erlandson s work on satisficiability What is a Filter? Notes from DSPDMS 1996, July; p. 91 (Coded on May 19, 2011): In very broad set theoretic terms, to filter is to separate, to partition a given set of entities into disjoint subsets, based on whether they have or do not have a stipulated property. One could, in principle, conceive of a Boolean Genie, as it were, that could instantly appear, on our rubbing a Lamp, and as instantly carry out the partitioning all the entities all at once. In the absence of such a Lamp, we must be content with a more earthly procedure, one that is a function of time and space, and which depends for its realization on a physical device, be it a sieve, an electronic circuit, or a computer. References [Thron] Wolfgang J. Thron, Topological Structures, Holt-Rinehart, 1966; iitk [Bushaw] D. Bushaw, Elements of General Topology, John Wiley, 1963; iitk [Steen] L. A. Steen, Highlights in the History of Spectral Theory, American Mathematical Monthly 80 (1973)

7 [Erlandson] R.F. Erlandson, The Satisfaction Process: Topology, Information Science 26, 1 43 (1982). A Characterization Using Extended [Simon] H. A. Simon, The Science of the Artificial, MIT Press, Cambridge, Mass., 1969.