RAMSEY ALGEBRAS AND RAMSEY SPACES

Transcription

1 RAMSEY ALGEBRAS AND RAMSEY SPACES DISSERTATION Presented in Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy in the Graduate School of the Ohio State University By Wen Chean Teh, M.S. Graduate Program in Mathematics The Ohio State University 2013 Dissertation Committee: Professor Timothy Carlson, Advisor Professor Chris Miller Professor Neil Robertson

2 Copyright by Wen Chean Teh 2013

3 ABSTRACT Hindman s theorem says that every finite coloring of the natural numbers has a monochromatic set of finite sums. Galvin and Glazer gave a brilliant simple proof of Hindman s theorem using idempotent ultrafilters. We study Ramsey algebras, which are structures that satisfy an analogue of Hindman s theorem. We show the existence of idempotent ultrafilters for Ramsey algebras under Martin s axiom, and the existence of idempotent ultrafilters for Ramsey algebras on a countable field of sets. We conclude by studying a class of Ramsey spaces, which arise from Ramsey algebras. ii

4 To my family, especially my wife Chin Yee. iii

5 ACKNOWLEDGMENTS I am grateful to Timothy Carlson, my advisor, for his support, patience, and encouragement. At tough times, he knows precisely when to lend a hand. His wise guidance prepared me to do independent research in my future academic career. He always finds the best in me. I learned a great deal from him, not just knowledge but wisdom in life. I would like to thank my committee members Chris Miller and Neil Robertson for their time and flexibility. I also want to thank them for providing valuable advice in the preparation of this dissertation. I am thankful to Cindy Bernlohr, who has provided a lot of assistance, be it related to my teaching assignment or not; Donald Robertson, my badminton partner, for proofreading this dissertation and providing valuable suggestions; and Guan Aun How, without whom I would not be in this beautiful country. I would like to thank The Ohio-State University and the Malaysia University of Sciences for their contribution to my education, which made me a better human being. Finally, I offer my gratitude to the dozens of professors, staff, colleagues, and friends who were there for me when I needed support. iv

6 VITA B.Sc. in Mathematics, Malaysia University of Sciences, Malaysia M.Sc. in Mathematics, Malaysia University of Sciences, Malaysia 2004 Present Graduate Teaching Associate, The Ohio State University FIELDS OF STUDY Major Field: Mathematics Specialization: Mathematical Logic and Combinatorics v

7 TABLE OF CONTENTS Abstract Dedication Acknowledgments Vita ii ii iv v Basic Terminology Basic Terminology and Notation L-Algebras and Terms CHAPTER PAGE 1 Introduction and Preliminaries Hindman s Theorem Ramsey Algebras Idempotent Ultrafilters Basic Facts Regarding Idempotent Ultrafilters Ramsey Algebras in Relation to Idempotents Ellentuck s Theorem Ramsey Spaces Basic Facts Regarding Ramsey Spaces Ramsey Spaces from Ramsey Algebras Classification of Various Algebras Unary Operations Associative Operations Non-associative Operations Rings Strongly Reductible Ultrafilters vi

8 4 Idempotent Ultrafilters on Fields of Sets Framework Existence of Idempotents Ultrafilters Algebras with Idempotents are Ramsey Generalized Ramsey Algebras and Spaces Bibliography vii

9 BASIC TERMINOLOGY Basic Terminology and Notation In this section, we present the terminology and notation used in this thesis. We for the most part follow those used in [3]. The natural numbers are defined so that n = {0,..., n 1} whenever n is a natural number. In particular, 0 is the empty set. The set of natural numbers {0, 1,... } is denoted by ω. The cardinality of a set is denoted by X. The cardinality of the continuum is denoted by c. Tuples are defined inductively using ordered pairs. There is a unique 0-tuple ( ) defined to be. The 1-tuple (x) is defined to be x. The 2-tuple (x, y) is the ordered pair of x and y. If n 2, then (x 0,..., x n ) is the ordered pair ((x 0,..., x n 1 ), x n ). 1 The Cartesian product A B of two sets A and B is the set of ordered pairs (a, b) where a A and b B. Products of sets are to be associated to the left unless indicated otherwise by parentheses, for example, A B C abbreviates (A B) C. The n-fold Cartesian product of A is denoted by A n. All functions are assumed to have a codomain. If a function is defined without an explicit mention of its codomain, then the codomain is assumed to be the range. A function f A B is one-to-one if a = b whenever f(a) = f(b), onto if the range 1 (x 0,..., x n ) is defined in [3] to be the ordered pair (x 0, (x 1,..., x n )). The difference is just a matter of convenience for our treatment in chapter 4. 1

10 of f is the codomain B and a bijection if it is both one-to-one and onto. The set of all functions with domain A and codomain B is denoted by A B. Suppose f A B, X A, and Y B. The set { f(x) x X } is denoted by f[x] and { a A f(a) Y } is denoted by f 1 [Y ]. The restriction of f to X with codomain B is denoted by f X. Following the notational convention in [3], we will use a symbol with a bar over it to indicate a list, e.g. given functions f, g and h with appropriate domains and codomains, we may define a function F by writing F ( x, ȳ) = f(g( x), h(ȳ)) for all x and ȳ, rather than F (x 1,..., x n, y 1,..., y m ) = f(g(x 1,..., x n ), h(y 1,..., y m )) whenever (x 1,..., x n ) is in the domain of g and (y 1,..., y m ) is in the domain of h. A finite sequence is a function whose domain is a natural number. An infinite sequence is a function with domain ω. We will simply say sequence if it is clear from the context whether the sequence is finite or infinite. An arrow above the name of an object will often be used to indicate it is a sequence, either finite or infinite. If s is a sequence and i is in the domain of s, then s(i) is sometimes called the i-th term of s. There will be no fixed relationship between the object denoted by a symbol and the object denoted by the same symbol with an arrow above it, e.g. s and s have no special fixed connection with each other. The finite sequence whose value at i is x i for i < n is denoted by x 0,..., x n 1. The empty sequence is denoted by. Let x 0, x 1,... or x i i ω denote the infinite sequence whose value at i is x i for all i ω. Note that the domain of a sequence (finite or infinite) is the same as its cardinality. Hence, s is the domain of a sequence, also called its length. If s = s 0,..., s n and t = t 0,..., t m, then the concatenation of s and t, denoted by s t, is s 0,..., s n, t 0,..., t m. Similarly, if s = s 0,..., s n and t = t 0, t 1,..., then s t is s 0,..., s n, t 0, t If n 1 and s is a sequence such that s n, then s n = s 0,..., s n 1. Let s 0 be the empty sequence. For n ω, let s n be the 2

11 sequence t satisfying ( s n) t = s. Initial segment and subsequence will have their usual meaning. To say that A i (i I) is an indexed collection means that A i (i I) is a function with domain I. The value of A i (i I) at j is denoted by A j. The notation {A i } i I means the same thing. If A i is a set for every i I, we say that A i (i I) is an indexed collection of sets. Suppose A is any set. P(A) denotes the power set of A. The collection of all finite subsets of A is denoted by P f (A). The collection of subsets of A of size ω is denoted by [A] ω. Meanwhile, <ω A denotes the collection of all finite sequences of elements from A. For n ω, n A denote the set of finite sequences in A of length n and [A] n denote the collection of subsets of A of size n. If f is an n-ary operation on a set A and a n A, we will sometimes write f( a) for f( a(0),..., a(n 1)) for notational convenience. If A is a set and n ω, an n-ary operation on A is a function with domain A n and codomain A. An operation on A is an n-ary operation for some n. Suppose f is an n-ary operation on A, and B A. We say that B is closed under f iff f[b n ] B. When the context indicates that B is closed under f, we sometimes abuse notation and write f B for the restriction of f to B n with codomain B; so f B is an operation on B. If F is a collection of operations on a nonempty set A, then (A, F) is called an algebra. If F is finite, we will write (A, f 1,..., f n ) instead of (A, {f 1,..., f n }), e.g. (ω, +, ). If B A and B is closed under f for every f F, then we write F B for { f B f F } and call (B, F B) a subalgebra of (A, F). An algebra (A, f) is a groupoid if f is a binary operation and is a semigroup if in addition f is associative. Suppose (A, f) and (B, g) are algebras where both f and 3

12 g are n-ary and that π A B. Then π is a homomorphism iff π(f(x 1,..., x n )) = g(π(x 1 ),..., π(x n )) for all x 1,..., x n A. A pre-partial ordering on a set A is a binary relation on A which is reflexive and transitive. If is also antisymmetric, it is called a partial ordering on A. Suppose A is a nonempty set. An ultrafilter on A is a collection U of subsets of A such that A U and U; if X, Y U, then X Y U; and for every X A, either X U or A/X U. Only elementary topological knowledge is needed in this thesis. The most basic texts in point set topology contain the relevant results. A set in a topological space has the property of Baire if and only if its symmetric difference with some open set is meager. L-Algebras and Terms We assume the reader is familiar with the syntax and semantics of first order logic. Standard undergraduate reference are [11], [26] and my personal recommendation is [9]. A casual reading through section 2.1 and 2.2 of [11] is more than sufficient. We do not need relation symbols or constant symbols. Hence, we identify a language L with its set of function symbols, none of which is nullary. An L-algebra A is simply an L-structure ( A, {f A } f L ) in the usual sense. There is a fixed list of variables v 0, v 1, v 2,.... The index of v i is i. A term is defined as usual. We will not use formulas. Suppose A is an L-algebra. An assignment has the usual meaning and will be identified with a sequence a ω A. Suppose t is a term of L. The interpretation of the term t under A with assignment a is as usual and is denoted by t A [ a], that is, the function symbols in t are interpreted according to A and every variable v i is interpreted as a(i). Unless otherwise stated, whenever we say t(v i0,..., v ip ) is a term, we implicitly 4

13 assume that i 0 < < i p, and that the variables appearing in t are among those shown. If t(v i0,..., v ip ) is a term, then t A [v i0,..., v in a i0,..., a in ] denote the interpretation of the term t under A with v ip being interpreted as a ip for each 0 p n. Every algebra (A, F) can be regarded as an L-algebra for the canonical language L. To be more explicit, let L = { f f F }, where f is a function symbol with the same arity as f. Hence, A = (A, F) is an L-algebra with universe A = A and f A = f for all f L. In this context, we assume the L is understood. 5

14 CHAPTER 1 INTRODUCTION AND PRELIMINARIES 1.1 Hindman s Theorem In Ramsey theory, Schur s theorem [34] states that for every finite partition of the positive integers, one of the cells contains two numbers (not necessarily distinct) and their sum. In the 1960 s, Folkman s theorem was obtained independently by Folkman (unpublished), Rado [29], and Sanders [33]. This result, which is a strong generalization of Schur s theorem, states that for every finite coloring of the natural numbers and every m ω, there exists a set H of size m such that { i F P(H) } i F is monochromatic. Graham and Rothschild [15] conjectured that the infinite version of this result was also valid. The infinite version had been an open problem for decades, even though several mathematicians (including Hilbert) had worked on it. 1 Hindman proved it in Suppose x i i ω is a sequence of natural numbers. Let FS( x i i ω ) denote the set { x i F P f (ω) }. i F Theorem (Hindman [18]). Suppose ω is finitely colored. Then there is a sequence x i i ω of natural numbers such that FS( x i i ω ) is monochromatic. Baumgartner ([2]) gave a relatively simple combinatorially proof of Hindman s 1 The claim that Hilbert had worked on this problem can be found in the preface of [23]. 6

15 theorem. He proved a version of Hindman s theorem that deals with finite unions of finite subsets of ω rather than finite sums, which was already known to be equivalent to Hindman s theorem. This version is sometimes called the finite union theorem while Hindman s theorem is also called the finite sums theorem. Because of its historical importance, we also state the finite union theorem. Suppose H i i ω is a sequence in P f (ω). Let FU( H i i ω ) denote the set { H i F P f (ω) }. i F Theorem (Hindman, [18]). Suppose P f (ω) is finitely colored. Then there is a sequence of pairwise disjoint sets H i i ω in P f (ω) such that FU( H i i ω ) is monochromatic. The combinatorial conclusion in Hindman s theorem can be strengthened and generalized to arbitrary semigroups. Let (S, ) be a semigroup and x i i ω, y i i ω ω S. For every H P f (ω), let x t denote the product in increasing order of indices. t H Bracketing is unnecessary because of associativity. Let FP( x i i ω ) denote the set { x t F P f (ω) }. We say that y i i ω is a product subsystem of x i i ω if and only t F if there is a sequence H i i ω in P f (ω) such that for every i ω, we have max H i < min H i+1 and y i = t H i x t. Theorem Suppose (S, ) is a semigroup, x i i ω ω S, and that S is finitely colored. Then there is a product subsystem y i i ω of x i i ω such that FP( y i i ω ) is monochromatic. Proof. See corollary 5.15 in [23]. 2 There are several places in this dissertation where we quote results that appear to be natural generalizations of the more well-known results in the literature. We do not know who the results should be attributed to. When we do not know of other references, we will refer the reader to the standard textbook [23] for these results. 7

16 1.2 Ramsey Algebras In Theorem 1.1.3, we see that semigroups have a certain combinatorial property. In this section, we will define the class of Ramsey algebras, which contains the class of semigroups. The notion of a Ramsey algebra was proposed by Carlson. Before describing it we generalize the notion of a product subsystem to a collection of operations. Definition Suppose F is a collection of operations on a set A. An operation f A n A is an orderly composition of F iff there are g, h 1,..., h n F such that f( x 1,..., x n ) = g(h 1 ( x 1 ),..., h n ( x n )). We say that F is closed under orderly composition iff f F whenever f is an orderly composition of F. The collection of orderly terms over F is the smallest collection of operations on A that contains F and the identity function on A and is closed under orderly composition. Definition Suppose F is a collection of operations on a set A and a, b are infinite sequences in A. We say a is a reduction of b, and write a F b iff there are finite sequences b k and orderly terms f k over F for all k ω such that b 0 b 1 b 2 is a subsequence of b and a(k) = f k ( b k ) for all k ω. It is easy to check that F is a pre-partial ordering on the collection of infinite sequences in A. Our definition of F is equivalent to a special case of the one given in [3], where the collection of operations contains all projections. If F = {f}, we will write f for F. The same convention will apply in future to similar situations without mention. Definition Suppose F is a collection of operations on a set A and b is an infinite sequence in A. An element a of A is a finite reduction of b iff a is equal to c(0) for some c F b. Define FRF ( b) to be the set of all finite reductions of b. 8

17 We usually write FR( b) for FR F ( b) when F is clear from the context. Note that FR( b) is the same as the set of all f( b 0 ) for which f is an n-ary orderly term over F and b 0 is a finite subsequence of b of length n. For example, when A = ω and F = {+}, the sets FR( b) and FS( b) are the same. Definition An algebra (A, F) is Ramsey iff for every X A and every a ω A, there exists b F a such that FR F ( b) is either contained in or disjoint from X. We say that b is homogeneous for X if and only if FR( b) is either contained in or disjoint from X. Later in this chapter, we will see that Ramsey algebras are closely related to Ramsey spaces (to be defined) and most algebras are Ramsey if and only if their corresponding spaces are Ramsey. Example If F =, then the empty algebra (A, F) is trivially Ramsey. Note that in this case, for every a, b ω A, we have a F b if and only if a is a subsequence of b. The Ramsey property follows by the pigeonhole principle. Theorem Every semigroup is a Ramsey algebra, so is every group. Proof. By Theorem It is easy to see that every subalgebra of a Ramsey algebra is Ramsey. We do not know if the Cartesian product of two Ramsey algebras is Ramsey. The following characterization of finite Ramsey algebras is an unpublished observation by Carlson. To say that a A is an idempotent element for an algebra (A, F) means that for every f F if f is n-ary, then f(a,..., a) = a. n times Theorem (Carlson). A finite algebra is Ramsey if and only if every subalgebra contains an idempotent element. 9

18 Proof. Fix a finite algebra (A, F). Note that for every x, y A, we know that y is in the subalgebra generated by x if and only if y FR( x, x, x,... ). ( ) Suppose B is a subalgebra. Fix an element x B. Since the algebra is finite, there is a finite coloring of A such that every element of A gets a unique color. Since (A, F) is a Ramsey algebra, there exists a sequence a ω A such that a F x, x, x... and FR( a) is monochromatic. By the choice of our coloring, FR( a) must consist of a single element, say y. Then a must be equal to y, y, y,... and so y is idempotent with respect to F. Since y FR( x, x, x... ) and the subalgebra generated by x is contained in B, it follows that y B. ( ) It suffices to show that for every sequence a in A, there is a sequence b F a such that FR( b) consists of a single element. By going to a subsequence, we may assume that the sequence a is x, x, x,... for some x A. Choose an idempotent element y from the subalgebra generated by x. Take b to be the sequence y, y, y,.... Clearly FR( b) = {y} because y is idempotent. Since y FR( x, x, x,... ), it is easy to see that b F a. Example Theorem fails trivially for infinite algebra. The semigroup (ω/{0}, +) is Ramsey but does not contain any idempotent elements. On the other hand, every subalgebra of (Z, ) contains an idempotent element, namely 0, but (Z, ) itself is not a Ramsey algebra (to be shown later in Theorem 2.3.2). Now, we introduce a natural class of algebras properly containing the class of Ramsey algebras. Definition An algebra (A, F) is weakly Ramsey iff for every X A, there exists b ω A such that FR F ( b) is either contained in or disjoint from X. Clearly every Ramsey algebra is weakly Ramsey and every algebra that has a weakly Ramsey subalgebra is weakly Ramsey. 10

19 Example It is easy to construct a weakly Ramsey algebra that is not Ramsey. Let f be any associative operation on the set of even numbers. We define an operation f(x, y) if x, y 2ω g on ω by g(x, y) =. Trivially (ω, g) is a weakly Ramsey algebra x + 2, otherwise because it has a Ramsey subalgebra, namely (2ω, f). We claim that (ω, g) is not a Ramsey algebra. Suppose not. Take x i i ω to be any sequence of odd numbers and X = 4ω + 1. Then there exists y i i ω g x i i ω homogeneous for X. It is easy to see that every y i is odd. Therefore, y 0 X if and only if g(y 0, y 1 ) = y X. Hence, neither FR( y i i ω ) X nor FR( y i i ω ) X c can be true, a contradiction. 1.3 Idempotent Ultrafilters Galvin observed that Hindman s theorem would follow from the existence of an ultrafilter U on ω such that { x ω A x U } U whenever A U, where A x = { y ω x + y A }. Galvin called such ultrafilters almost translation invariant ultrafilters. Hindman [17] showed the existence of such ultrafilter assuming the continuum hypothesis before he proved the Hindman s theorem. In 1975 Glazer (unpublished) proved the existence of such ultrafilter without continuum hypothesis. 3 He defined an operation + on βω, the Stone-Čech compactification on ω, which extends ordinary addition on ω as follows. For every U, V βω (that is, U and V are ultrafilters on ω), U + V = { X ω { x { y x + y X } V } U }. 3 The above historical account is taken from [21]. 11

20 The main observation was that an almost translation invariant ultrafilter is exactly an idempotent element with respect to +. As (βω, +) is a compact right topological semigroup, it has an idempotent element. See sections 7 and 8 of [19] for a presentation of the Stone-Čech compactification and Glazer s proof. Next, we give a proof of the observation by Galvin for motivation. Theorem (Galvin). Suppose U is an ultrafilter on ω such that U + U = U. If X U, then there exists b ω ω such that FS( b) X. Proof. For brevity, in this proof, we assume x and y run through natural numbers. U + U = U means that U = { X ω { x { y x + y X } U } U }. Suppose X U. Then { x { y x + y X } U } U. We construct a sequence b i i ω ω ω and a sequence of sets X i i ω ω U inductively as follows. Let X 0 be X. Choose b 0 to be any element of X 0 { x { y x + y X 0 } U }. Then { y b 0 + y X 0 } U. Let X 1 = X 0 { y b 0 + y X 0 }. Inductively, suppose X n U has been constructed. Since U is idempotent, { x { y x + y X n } U } U. Choose b n to be any element of X n { x { y x + y X n } U }. Let X n+1 = X n { y b n + y X n }. Note that X 0 X 1 X 2. By the definition, FS( b i i ω ) X follows if we show that for every n ω, whenever i 0 < i 1 < < i n, we have b i0 + b i1 + b in X i0. We will prove this by induction on n. Clearly the base case holds trivially: b i0 X i0 for all i 0 ω. Now, for the inductive step, by the induction hypothesis, b i1 + + b in+1 X i1. Since X i1 X i0 +1 = X i0 { y b i0 + y X i0 }, it follows that b i1 + + b in+1 { y b i0 + y X i0 }. Thus b i0 + b i1 + + b in+1 X i0. The argument in the Galvin-Glazer proof of Hindman s theorem works for any semigroup. So, if (S, ) is a semigroup, then there exists an ultrafilter U on S such 12

21 that U U = U, where U U = { X S { x { y xy X } U } U }. If U U = U and X U, then there exists b ω S such that FP( b) X. Now, we present a generalization of the Galvin-Glazer approach to a collection of operations. This generalization appears in [3]. Definition Assume U is an ultrafilter on A and f A B. Let f (U) be the ultrafilter on B defined by { X B f 1 [X] U }. If f is a bijection, we say that U and f (U) are isomorphic and f is an isomorphism between U and f (U). For notational convenience, we will write f(u) for f (U). It is easy to check that f(u) is indeed an ultrafilter on B. This manner of projecting ultrafilters has been studied extensively (see [7]). Definition Assume U and V are ultrafilters on sets A and B respectively. U V is the ultrafilter on A B defined to be { X A B { a A { b B (a, b) X } V } U } It is easy to check that U V is indeed an ultrafilter on A B. Products of ultrafilters are to be associated to the left unless stated otherwise by parentheses, for example, U V W abbreviates (U V ) W. This is inessential because (U V ) W and U (V W ) are isomorphic. Let U n denote the n-fold product of U. Suppose U i is an ultrafilter on A i for i = 1,..., n and f A 1 A n B. We will write f(u 1,..., U n ) for f(u 1 U n ). Definition Suppose F is a collection of operations on a set A. An ultrafilter U on A is said to be idempotent for F iff f (U n ) = U whenever f F and f is n-ary. If f is an operation on A, we will write U is idempotent for f rather than U is idempotent for {f}. Note that if f is binary and U is idempotent for f, then X U if and only if { a 1 A { a 2 A f(a 1, a 2 ) X } U } U. 13

22 Suppose A is a set and a A. Suppose U A a denote the principal ultrafilter on A generated by a, that is, U A a = { X A a X }. We will usually write U a for U A a when A is understood. For every a 1,..., a n A, we have Ua A 1 Ua A n = U An (a 1,...,a n). If f is an n-ary operation on A, then f (U a1,..., U an ) is equal to U f(a1,...,a n). Hence, f can be regarded as an extension of f if the elements of A are identified with the principal ultrafilters on A. Furthermore, U a is idempotent for f if and only if f(a,..., a) = a. n times Suppose U is an ultrafilter on A and X A n. Then we can check inductively that for every 1 k n 1, we have X U n if and only if { (a 1,..., a k ) A k { (a k+1,..., a n ) A n k (a 1,..., a n ) X } U n k } U k. Suppose F is a collection of operations on a set A. The following property of F allows us to carry out the construction of a homogeneous sequence assuming F has an idempotent ultrafilter. Definition Assume A is a set and U is an ultrafilter on A. Suppose F is a collection of operations on A. Let F(m) denote the set of m-ary operations in F. We say that F is finitely based if for each m 1, F(m) is finite; There exists a finite subcollection F of F such that for each F F(m + l) with l > 0, there are n, l 0,..., l n ω where l = l l n such that for some f F (m + n) and g k F(l k ), F is given by F (ā, x 0,..., x n ) = f(ā, g 0 ( x 0 ),..., g n ( x n )). Theorem (Carlson). Assume F is a collection of operations on a set A and U is an ultrafilter on A idempotent for F. If F is finitely based, then for every X U, there exists a ω A such that FR( a) X. Proof. See Lemma 3.10 in [3]. 14

23 1.4 Basic Facts Regarding Idempotent Ultrafilters The following basic facts about projecting ultrafilters allow us to show that if U is idempotent for a collection of operations F, then it is idempotent for the collection of orderly terms over F. Proposition (Carlson [3]). Assume U is an ultrafilter on A. If f A B and g B C, then (g f)(u) = g(f(u)). Proposition (Carlson [3]). Assume f i A i B i for i = 1,..., n and G A 1 A n B 1 B n is given by G(a 1,..., a n ) = (f 1 (a 1 ),..., f n (a n )). If U i is an ultrafilter on A i for i = 1,..., n, then G(U 1,..., U n ) = f 1 (U 1 ) f n (U n ). Theorem (Carlson). Suppose F is a collection of operations on a set A. If U is an ultrafilter on A idempotent for F, then it is idempotent for the collection of orderly terms over F. Proof. By Propositions and or see Lemma 3.7 in [3]. The following four propositions are natural properties of idempotent ultrafilters. We do not know whether they have appeared in the literature or who should they be credited to. We give them without proofs. Proposition Suppose F is a collection of operations on a set A and B is a subset of A such that B is closed under f for every f F. Assume U is an ultrafilter on B. Let V be { X A X B U }. If U is idempotent for F B, then V is an ultrafilter on A idempotent for F. 15

24 Proposition Suppose F is a collection of operations on a set A and B is a subset of A such that B is closed under f for every f F. Assume V is an ultrafilter on A such that B V. Let U be { X B X V }. If V is idempotent for F, then U is an ultrafilter on B idempotent for F B. Proposition Suppose f and g are n-ary operations on A and B respectively, that π A B and that U is an ultrafilter on A. Assume π is a homomorphism or more generally { (a 1,..., a n ) A n π(f(a 1,..., a n )) = g(π(a 1 ),..., π(a n )) } U n. If U is idempotent for f, then π(u) is idempotent for g. Proposition Suppose f and g are n-ary operations on a set A and U is an ultrafilter on A. If { (a 1,..., a n ) A n f(a 1,..., a n ) = g(a 1,..., a n ) } U n and U is idempotent for f, then U is idempotent for g. Suppose f and g are binary operations on a set A such that each has an idempotent ultrafilter. We do not know whether the Cartesian product h A 2 A 2 A 2 defined by h( (a, b), (c, d) ) = (f(a, c), g(b, d)) has an idempotent ultrafilter. Note that if U and V are idempotent for f and g respectively, U V need not be idempotent for h. 1.5 Ramsey Algebras in Relation to Idempotents By Theorem 1.2.7, we know that a finite algebra is Ramsey if and only if every subalgebra has an idempotent ultrafilter. We will discuss this type of connection for infinite algebras in this section. Theorem (Carlson). Assume (A, F) is an algebra and that the collection of orderly terms over F is finitely based. If there exists an ultrafilter idempotent for F, then (A, F) is a weakly Ramsey algebra. Proof. Suppose U is idempotent for F. Then by Lemma 1.4.3, it is idempotent for the collection of orderly terms over F. Apply Theorem

25 Note that if F = {f} and f is a semigroup operation, then the collection of orderly terms over F is finitely based. Although idempotent ultrafilters for semigroups have been studied extensively (see [23]), the same cannot be said about idempotent ultrafilters for general algebras. We do not know of an example of an algebra with idempotent ultrafilters that is not finitely based and thus that is not necessarily a weakly Ramsey algebra. Suppose (A, ) is a semigroup and a ω A. The Galvin s method of constructing homogeneous sequences can be refined so that the homogeneous sequences are reductions of a. This is made possible by the next theorem. Theorem (Galvin). Suppose (A, ) is a semigroup and a ω A. Then there is an ultrafilter on A idempotent for such that FP( a m) U for all m ω. Proof. See Lemma 5.11 in [23]. Theorem Suppose (A, ) is a semigroup, a ω A, and U is an ultrafilter on A idempotent for such that FP( a m) U for all m ω. If X U, then there exists b ω A such that b { } a and FP( b) X. Proof. See Theorem 5.14 in [23]. Theorem is generalized by Carlson to a collection of finitely based operations and the following is an easy corollary. Theorem (Carlson [3]). Assume (A, F) is an algebra and the collection of orderly terms over F is finitely based. Suppose for every a ω A, there exists an ultrafilter U idempotent for F such that FR( a m) U for all m ω. Then (A, F) is a Ramsey algebra. 17

26 Example Like example , it is easy to construct an algebra with idempotent ultrafilters that is not Ramsey. Suppose (2ω, f) is a semigroup. Define f(x, y) g(x, y) = x + 2, if x, y even. otherwise We see from example that g is not a Ramsey algebra. Meanwhile, by Proposition 1.4.4, an ultrafilter on 2ω idempotent for f induces an ultrafilter on ω idempotent for g. Question. Does every Ramsey algebra have idempotent ultrafilters? We do not know the answer to the above interesting question posed by Carlson. A positive answer will be a significant improvement on the result that every semigroup has idempotent ultrafilters. In chapter 3, we will show that assuming some special set theory axioms, every Ramsey algebra has an idempotent ultrafilter with some stronger property. 1.6 Ellentuck s Theorem Suppose A is any set. Let [A] ω denote the collection of all subsets of A of size ω. [ω] ω can be naturally embedded into the Cantor space ω 2. Hence, [ω] ω has the induced subspace topology where ω 2 has the Tychonoff product topology. A set X [ω] ω is called Ramsey if and only if there exists H [ω] ω such that [H] ω is either contained in or disjoint from X. Erdös and Rado [12, Example 1, p.434] showed that not every X [ω] ω is Ramsey. Their counterexample is nonconstructive. The classical Ramsey theorem [30] can be regarded as the first positive result on the classification of subsets of [ω] ω that are Ramsey. In 1968, Nash-Williams [28] proved a result that is halfway between clopen sets are Ramsey and open sets are Ramsey. 18 This result is a generalization of

27 Ramsey s theorem as for k ω, every 2-coloring of [ω] k induces a clopen set in [ω] ω. Galvin [13] proved that open sets are Ramsey. Galvin and Prickry [14] showed based on the method of Nash-Williams that every Borel subset of [ω] ω is Ramsey. Silver [36] later showed that every analytic subset of [ω] ω is Ramsey. In 1974, Ellentuck [10] gave a new proof of Silver s result by introducing a finer topology on [ω] ω, which we will call the Ellentuck topology. The space [ω] ω with this topology will be called the Ellentuck space. His proof is based on the method of Galvin-Prikry [14]. For every n ω and A [ω] ω, let [n, A] ω = { B [ω] ω B A and B contains the first n elements of A }. The Ellentuck topology on [ω] ω is the topology generated by the sets [n, A] ω. A set X [ω] ω is called completely Ramsey if and only if for every n ω and A [ω] ω, there exists B [n, A] ω such that [n, B] ω is either contained in or disjoint from X. Theorem (Ellentuck [10]). Suppose X [ω] ω. Then X is completely Ramsey if and only if X has the property of Baire under the Ellentuck topology. Since every analytic subset of [ω] ω has the property of Baire under the Ellentuck topology, every analytic set is completely Ramsey. 1.7 Ramsey Spaces In 1988, Carlson [3] presented a framework which generalizes the results of Ellentuck. Carlson called structures that satisfy an analogue of Ellentuck s theorem Ramsey spaces. His abstract result reduces the topological question of whether such a space is Ramsey to a more combinatorial question. We will now present Carlson s definition of Ramsey spaces and results of his that are relevant to our work. Most definitions are presented as appeared in [3]. 19

28 Definition Assume R is a nonempty set, p is a function with domain ω R, is a pre-partial order on R, and R = (R,, p). R is a pre-partial order with approximations if the following assumptions hold. p(0, A) = p(0, B), for all A, B R. If A and B are distinct elements of R, then p(n, A) p(n, B) for some n ω. If p(m, A) = p(n, B), then m = n and p(i, A) = p(i, B) for all i < n. R is the universe of R. p(n, A) is the n-th approximation of A and the depth of p(n, A) is n. β is an approximation of R iff β = p(n, A) for some A R, and n ω. If A, B R and A B, then A is called a reduction of B with respect to R. If R satisfies (1) (3) and in addition is a partial order on R, then R is a partial order with approximations. Suppose R = (R,, p) is a pre-partial order with approximations. Some additional notations will be useful. For A R, let p(a) denote the sequence of approximations of A: p(a)(n) is the n-th approximation of A. If R is a set of infinite sequences, p is the usual restriction function given by p(n, A) = A n and is a pre-partial order on R, then R will be identified with (R, ). Fix a pre-partial order with approximations R = (R,, p) for the rest of the following definitions. Definition Assume P is the set of approximations of R. The product topology on R is the topology induced by the embedding of R into ω P which sends A to p(a), where P is given the discrete topology. R is closed if { p(a) A R } is a closed subset of ω P. Note that the collection of sets of the form { A R α is an approximation of A }, where α is an approximation is a base for the product topology. 20

29 Remark. In the case that R is a set of infinite sequences in some set S, R being closed is equivalent to R being a closed subset of the product space ω S, where S is given the discrete topology. Definition If β is an approximation and A R, define R(β, A) to be the set of reductions of A which have β as an approximation. R(n, A) denotes R(p(n, A), A), for n ω and A R. The natural topology on R is the topology generated by the sets R(n, A). Notice that the sets R(n, A) for n ω form a neighborhood basis for A in the natural topology and that the natural topology contains the product topology. Unless otherwise stated, future discussions of topological matters will refer to the natural topology. Definition Assume X is a subset of R. We say that X is Ramsey 4 in R iff for every n ω and A R, there exists B R(n, A) such that R(n, B) is either contained in or disjoint from X. Define X to be Ramsey null iff for every n ω and A R, there exists B R(n, A) such that R(n, B) is disjoint from X. We say that R is a Ramsey space iff every set which has the property of Baire 5 is Ramsey and every meager set is Ramsey null. As remarked in [3], using the axiom of choice, one can show that if every set which has the property of Baire is Ramsey, then every meager set is Ramsey null. Since the study of Ramsey spaces in cases where the axiom of choice fails is also of some interest (see [5]), the clause requiring that each meager set is Ramsey null is included in the definition of a Ramsey space. 4 This notion corresponds to being completely Ramsey in Ellentuck s theorem. 5 The reader is reminded that a subset in a topological space has the property of Baire iff it differs from an open set by a meager set. 21

30 Definition Assume is a pre-partial order whose domain contains the set of approximations of R. Suppose α and β are approximations of R. To say that α is a reduction of β (with respect to ) simply means that α β. We say that is a finitization of R if whenever A, B R, A is a reduction of B iff every approximation of A is a reduction of an approximation of B. Let A1, A2 and A3 stands for the following properties. A1. Each approximation of R has only finitely many reductions. A2. If B R(β, A), where β is a reduction of p(n, A) but not of p(i, A) for i < n, then there is A R(n, A) such that R(β, A ) R(β, B). A3. For each n ω, if Z is a set of approximations of depth n + 1 and A R, then there exists B R(n, A) such that the set of all (n + 1)-th approximations of elements of R(n, B) is either contained in or disjoint from Z. Theorem (Abstract Version of Ellentuck s Theorem, Carlson, [3]). If R is closed and has a finitization satisfying A1 and A2, then R is a Ramsey space iff R satisfies A3. Note that if R is a Ramsey space, then A3 follows immediately because a set of approximations of a fixed depth induces a set that is open in the product topology, and hence open in the natural topology of R. Example Let E be the pre-partial ordering with approximation ([ω] ω,, p) such that for every n ω and A, B [ω] ω, p(n, A) is the set consisting of the first n elements of A and A B if and only if A B. Under the natural topology, E is precisely the Ellentuck space. Using the abstract version of Ellentuck s theorem, the fact that E is a Ramsey space reduces to the fact that E satisfies the combinatorial statement A3; the latter fact follows trivially from the pigeonhole principle. 22

31 Example Using Theorem 1.7.6, Carlson proved [3] that certain spaces of variable words with finite alphabets are Ramsey. These result have as corollaries many earlier Ramsey-theoretic results, including Ellentuck s theorem [10], the dual Ellentuck theorem [4], the Hales-Jewett theorem [16], and the Graham-Rothschild theorem on n-parameter sets [15]. The combinatorial statement A3 was proved by constructing idempotent ultrafilters. 1.8 Basic Facts Regarding Ramsey Spaces Suppose R = (R,, p) is a pre-partial order with approximations and S is a subset of R. We can easily check that S = (S, (S S), p (ω S)) is a pre-partial order with approximations. Also, S(n, A) = R(n, A) S for all n ω and A S. Hence, the natural topology on S is the same as the subspace topology on S induced from the natural topology on R. Our next example will show that a subspace of a Ramsey space need not be Ramsey. Example Recall the Ellentuck space E (see example 1.7.7). Let S be the collection of infinite subsets such that both the odd numbers and the even numbers occur infinitely often. We claim that the subspace S induced by S is not Ramsey. Let X be the set { A S the first element of A is odd }. Note that X = { S(1, A) A S and the first element of A is odd }; hence X is open in the natural topology on S. But for every A S, it is clear that S(0, A) is not homogeneous for X. Definition Suppose R = (R,, p) is a pre-partial order with approximations. For every n ω and A R, let R[n, A] be the triple (R(n, A),, p ) such that p (m, B) = p(n + m, B) for all m ω and B R(n, A). Note that for each m ω and B R(n, A), we have that ( R[n, A] )(m, B) = R(n + m, B) = R(n + m, B) R(n, A) because R(n + m, B) R(n, A). Hence, the 23

32 natural topology on R[n, A] is the subspace topology on R(n, A) induced by the natural topology on R. Furthermore, R is a Ramsey space if and only if R[n, A] is a Ramsey space for all n ω and A R. The following simple fact is known to Carlson but does not appear in the literature. Proposition (Carlson). Suppose R = (R,, p) is a pre-partial order with approximation. Suppose that for every A R and n ω, there exists B R(n, A) such that R[n, B] is a Ramsey subspace. Then R is a Ramsey space. Proof. Suppose A R, n ω and X R has the property of Baire. Choose B R(n, A) such that R[n, B] is a Ramsey subspace. Let R be R[n, B]. Then X R(n, B) has the property of Baire in the subspace topology of R. Hence there exists C R(n, B) such that R(0, C) is either contained in or disjoint from X R(n, B). Note that R(n, C) = R(0, C). Since R(n, C) R(n, B), it follows that R(n, C) is either contained in or disjoint from X. Definition Suppose R 1 = (R 1, 1, p 1 ) and R 2 = (R 2, 2, p 2 ) are pre-partial orders with approximations. The product R 1 R 2 = (R 1 R 2,, p) is defined by 1. p(n, (A 1, A 2 )) = (p 1 (n, A 1 ), p 2 (n, A 2 )) for all (A 1, A 2 ) R 1 R 2 and n ω; 2. (A 1, A 2 ) (B 1, B 2 ) if and only if A 1 1 B 1 and A 2 2 B 2 for every (A 1, A 2 ), (B 1, B 2 ) R 1 R 2. It is straightforward to verify that R is a pre-partial order with approximations. In fact, as spaces, R 1 R 2 is the product space of R 1 and R 2 ; in other words, the natural topology on R 1 R 2 is the product of the natural topologies on R 1 and R 2. Furthermore, R(n, (A 1, A 2 )) = R 1 (n, A 1 ) R 2 (n, A 2 ) for all (A 1, A 2 ) R 1 R 2 and n ω. However, R 1 R 2 need not be a Ramsey space when both R 1 and R 2 are. The following example shows this. 24

33 Example Suppose R = E E, where E is the Ellentuck space. We will identify every A [ω] ω with a strictly increasing sequence of natural numbers. Let X be the set { ( a, b) [ω] ω [ω] ω a(0) < b(0) }. Note that X = { R(1, ( a, b)) a(0) < b(0) and ( a, b) [ω] ω [ω] ω }; hence, X is open in the natural topology of R. But, for every ( a, b) [ω] ω [ω] ω it is clear that R(0, ( a, b)) is not homogeneous for X. Definition Suppose R 1 = (R 1, 1, p 1 ) and R 2 = (R 2, 2, p 2 ) are pre-partial orders with approximations and π R 1 R 2. We say that π is a homomorphism from R 1 into R 2 if π[r 1 (n, A)] = R 2 (n, π(a)) for all n ω and A R 1. We say that π is an isomorphism of R 1 and R 2 if it is a homomorphism and a bijection. Note that a triple (R,, p) is a pre-partial order with approximations if and only if it is isomorphic in a canonical way to some (S, ), where S is a set of infinite sequences and is a pre-partial ordering of S. Proposition (Carlson [3]). Assume R 1 and R 2 are pre-partial orders with approximations and π is a homomorphism of R 1 onto R 2. If R 1 is a Ramsey space, then so is R Ramsey Spaces from Ramsey Algebras The Ramsey spaces that we are interested in arise from some underlying Ramsey algebras. See [37] for other examples of Ramsey spaces. Recall the notion F that we introduced in section 1.2 (definition 1.2.2). Definition Suppose F is a collection of operations on a set A. Let R(A, F) denote the pair ( ω A, F ). 25

34 Since F is a pre-partial ordering on ω A, it follows that R(A, F) is a closed prepartial order with approximations. 6 Hence, we can endow R(A, F) with the natural topology and ask whether it is a Ramsey space or not. Note that for every n ω and a ω A, we have that the subspace ( R(A, F))[n, a] is isomorphic to ( R(A, F))[0, a n]. Therefore, R(A, F) is a Ramsey space if and only ( R(A, F))[0, a] is a Ramsey subspace for all a ω A. The abstract version of Ellentuck s theorem ties the property that R(A, F) is a Ramsey space to the property that (A, F) is a Ramsey algebra. In order to apply the abstract theorem to R(A, F), we need to have a pre-partial ordering on its set of approximations, that is, the set of finite sequences in A. Definition Suppose F is a collection of operations on a set A. Define the pre-partial order F on the set of finite sequences of elements of A by a F b iff there are finite sequences b k and orderly terms f k for k < a such that b 0 b a 1 is a subsequence of b and a(k) = f k ( b k ) for k < a. We say that a is a reduction of b iff a F b. Lemma (Carlson). Suppose F is a collection of operations on a set A. Then F is a finitization of R(A, F) satisfying property A2. Proof. See Lemmas 4.3 and 4.12 of [3]. Lemma (Carlson). Suppose F is a collection of operations on a set A. Then R(A, F) satisfies property A3 if and only if (A, F) is a Ramsey algebra. Proof. The proof of Lemma 4.14 in [3] shows this. Theorem (Carlson). Suppose F is a collection of operations on a set A. If F satisfies property A1, then R(A, F) is a Ramsey space if and only if (A, F) is a Ramsey algebra. 6 We remind the reader that for every a ω A, the n-th approximation of a is a n. 26

35 Proof. Suppose F satisfies property A1. By Lemma and the abstract version of Ellentuck s theorem, R(A, F) is a Ramsey space if and only if R(A, F) satisfies property A3. Apply Lemma Corollary Suppose F is any finite collection of operations, none of which is unary. Then R(A, F) is a Ramsey space if and only if (A, F) is a Ramsey algebra. Proof. By theorem since F satisfies property A1 trivially. Remark. If R(A, F) is a Ramsey space, then (A, F) is immediately a Ramsey algebra without the requirement that F satisfies property A1. This follows mainly because every subset of A induces a clopen set in R(A, F). Example The empty algebra (A, ) is trivially Ramsey by the pigeonhole principal. Thus R(A, ) is a Ramsey space by Theorem Similarly, R(A, f) is a Ramsey space for every semigroup (A, f). Milliken [27], using Hindman s theorem as a strong pigeonhole principle, proved that R(ω, +) is a Ramsey space before Carlson formulated the notion of Ramsey spaces. By Theorem 1.9.5, this follows easily. 27

36 CHAPTER 2 CLASSIFICATION OF VARIOUS ALGEBRAS In each section of this chapter, we will consider a certain class of algebras and obtain results that classify or partially classify its members according to the following criteria. 1. Does the algebra have idempotent ultrafilters? 2. Is the algebra Ramsey? If not, is it weakly Ramsey? As discussed in section 1.5, the answers to the above questions are related. 2.1 Unary Operations In this section, we will show that if we begin with a Ramsey algebra for which the operations are non-unary and expand it by adding unary operations, then the expanded algebra is Ramsey if and only every sequence has as reduction a sequence with a certain fixed points property. As a corollary, we obtain a classification of Ramsey algebras and Ramsey spaces for which the collection of operations are all unary. Lemma (Katetǒv [24]). Let A be any set and f A A an operation without fixed points. Then there is a partition A = A 1 A 2 A 3 such that A i f[a i ] = for i = 1, 2, 3. 28

37 Theorem Suppose F is a collection of unary operations on a set A and G is a collection of non-unary operations on A. Let S = { a A f(a) = a for all f F }. Then (A, F G) is a Ramsey algebra if and only if for every a ω A, there exists b F G a such that 1. FR G ( b) S; 2. for every X A, there exists c G b such that { d(0) d G c } is either contained or disjoint from X. Proof. ( ) Suppose a ω A and X A. Choose b F G a satisfying properties 1 and 2. Then choose c G b such that { d(0) d G c } is either contained in or disjoint from X. Clearly c F G a. It suffices to note that property 1 implies that { d(0) d G c } = { d(0) d F G c }. ( ) First of all, we define a useful coloring. Let α be a symbol not in A. By the axiom of choice, there exists an operation g A {α} A {α} such that 1. g does not have a fixed point; 2. if a S, then g(a) = α; and 3. if α a S, then there exists f F such that g(a) = f(a). By Lemma 2.1.1, there exists a partition A {α} = B 1 B 2 B 3 such that B i g[b i ] = for i = 1, 2, 3. This induces a partition A = A 1 A 2 A 3 such that for i = 1, 2, 3, whenever a A i S c, there exists f F such that f(a) A i. It suffices to show that for all a ω A, there exists b F G a such that c(0) S whenever c G b. We argue by contradiction. Suppose there exists a A ω such that for every b F G a, there exists c G b such that c(0) S. Fix such an a. Since (A, F G) is a Ramsey algebra, we can choose b F G a such that for some 1 i 3, if c F G b, then c(0) Ai. By assumption on a, we can choose c F G b such that 29

38 c(0) S. Then c(0) A i S c. This implies that f( c(0)) A i for some f F. It follows that d = f( c(0)) ( c 1) F G b but d(0) Ai, a contradiction. Corollary Suppose A, F, G, and S are as in Theorem Assume (A, G) is a Ramsey algebra. Then (A, F G) is a Ramsey algebra if and only if for every a ω A, there exists b F G a such that FR G ( b) S. Proof. Straightforward by Theorem Corollary Suppose A, F, G and S are as in Theorem Assume R(A, G) is a Ramsey space. Then R(A, F G) is a Ramsey space if and only if for every a ω A, there exists b F G a such that FR G ( b) S. Proof. ( ) Straightforward by Corollary and the fact that an algebra is Ramsey if the corresponding space is Ramsey. ( ) Let R denote R(A, F G). Suppose n ω and a ω A. By Proposition 1.8.3, it suffices to show that there exists b R(n, a) such that R[n, b] is a Ramsey subspace. By assumption, choose b 0 F G a n such that c(0) S whenever c G b0. Take b = a n b 0. Note that b R(n, a) and that R[n, b] is isomorphic to R[0, b 0 ]. Since c(0) S whenever c G b0, it follows that R[0, b 0 ] and R(A, G)[0, b 0 ] are the same spaces. By the hypothesis, R(A, G)[0, b 0 ] is a Ramsey subspace of R(A, G). Corollary Suppose F is a collection of unary operations on a set A. Let S = { a A f(a) = a for all f F }. Then the following are equivalent. 1. R(A, F) is a Ramsey space. 2. (A, F) is a Ramsey algebra. 3. For every a A, there exists b S such that b F a. 30