Patching Theorem and its applications

Bulletin of the Section of Logic Volume 34/3 (2005), pp. 143–149

Adam Kolany

´ SELECTION LEMMA AND OTHER RADO COMBINATORIAL STATEMENTS UNIFORMLY PROVED

Abstract We present an uniform method of proving some combinatorial statements of Rad´ o Selection Lemma type. They mainly come from Cowen [1], [2], [3].

Keywords: Rad´o Selection Lemma, Satisfiability, Compactness Theorem.

Patching Theorem and its applications Definition 1. Let C = {Ck S : k ∈ K} be a family of finite sets S and let R be a symmetric relation on k∈K Ck . A function G : K → k∈K Ck is a R-consistent choice on C iff 1. G(k) ∈ Ck , for every k ∈ K; 2. hG(k), G(l)i ∈ R, for different k, l ∈ K. L Ã o´s and Ryll-Nardzewski in [6] prove the following: Theorem 2. Let {Ck :Sk ∈ K} be a family of finite sets and let R be a symmetric relation on k∈K Ck . Suppose that for every finite K0 ⊆ K there exists an R-consistent choice on {Ck : k ∈ K0 }. Then there exists an R-consistent choice on {Ck : k ∈ K}.

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Using this we will prove: Theorem 3 (Patching Theorem). Let A = {Aj : j ∈ J} be a family of nonempty sets and let F = {Fj : j ∈ J} be such a family of nonempty finite sets of functions that Aj ⊆ dm(f ), for each f ∈ Fj , j ∈ J. If for every finite J0 ⊆ J there exists a function F0 with F0 dAj ∈ Fj , for j ∈ J0 , then there exists a function F with F dAj ∈ Fj , for each j ∈ J. Proof. Let K = J and Ck = {f ∈ FS k : dm(f ) = Ak }, k ∈ J, and let hf, gi ∈ R iff f ∪ g is a function, f, g ∈ k∈K Ck . We will prove that for every finite K0 ⊆ K, there exists an R-consistent choice on {Ck : k ∈ K0 }. Indeed, let K0 ⊆ K be finite. By the assumptions there exists a function F0 such that F0 dAk ∈ Fk , for k ∈ K0 . Then, however, {hk, F0 dAk i : k ∈ K0 } is an R-consistent choice on {Ck : k ∈ K0 }. Thus there exists an S R-consistent choice G on the whole {Ck : k ∈ K}. Then, however, F = k∈K G(k) is a function and F dAk = G(k)dAk = G(k) ∈ Ck ⊆ Fk , for any k ∈ K. 2 Let J0 ⊆ J. If F0 dAj ∈ Fj , for j ∈ J0 , we say that F0 patches {Aj : j ∈ J0 } wrt. F. Although patching is relative to a family F, we will often omit mentioning about this when it is obvious which F is intended. Now, the Patching Theorem can be restated as follows: If every finite subfamily A0 of A is patched by some F0 wrt. F, then the entire A is patched by some F wrt. F. To show how the Patching Theorem can be used, let us prove the well known Rado Selection Lemma. Theorem 4 (Rado Selection Lemma). Let {Ks | s ∈ S} be a family of finite subsets of X and let {γS0 : S0 ∈ F in(S)} be a family of functions such that γ(S0 ) : S0 → X and ∀s∈S0 (γ(S0 )(s) ∈ Ks ), for each finite S0 ⊆ S. Then there is a function f : S → X such that for every finite S0 ⊆ S, there exists such a finite S1 ⊆ S that satisfies S0 ⊆ S1 and γ(S1 )dS0 = f dS0 . Proof. Let J = Fin(S) and let AS0 = S0 and FS0 = {γ(S1 )dS0 : S0 ⊆ S1 }, for S0 ∈ Fin(S). We will show that finite subfamilies of A = {AS0 : S0 ∈ Fin(S)} are patched. So let S1 , . . . , Sn ⊆ S be finite and let F0 = γ(S1 ∪ . . . ∪ Sn ). Naturally, F0 dS1 ∈ FS1 , . . . , F0 dSn ∈ FSn , which means that F0 patches {S1 , . . . , Sn }. So every finite subfamily of A is

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patched, hence by the Patching Theorem, there exists F which patches the entire A. Then, however, f = F satisfies the thesis of the theorem. 2 Below we will formulate other combinatorial statements which were considered by R.H. Cowen in [1], [2] and [3]. We will show how they can easily be proven by using the Patching Theorem. A family of sets {XU0 : U0 ∈ Fin(U )} is decreasing iff XU1 ⊆ XU2 , for U2 ⊆ U1 , U1 , U2 ∈ Fin(U ). Theorem 5[1, Th. 1] Let H = {HI0 : I0 ∈ Fin(I)} be such a decreasing family of nonempty sets of partial functions on I that domains of functions in HI0 contain I0 , where I0 ∈ Fin(I). Moreover, let the set {h(i) : h ∈ H∅ } be finite, for each i ∈ I. Then, there is such a function F defined on I that for every finite I0 ⊆ I, there exists such h ∈ HI0 that hdI0 = F dI0 . Proof. Let J = Fin(I), AI0 = I0 and let FI0 = {hdI0 : h ∈ HI0 }, for I0 ∈ Fin(I). Let us take now I1 , . . . , In ∈ Fin(I) and let I0 = I1 ∪ . . . ∪ In . Since HI0 6= ∅, there is a function F0 ∈ HI0 . Of course I0 ⊆ dm(F0 ). Now, by monotonicity of H, we get F0 dI1 ∈ FI1 , . . . , F0 dIn ∈ FIn . That is F0 patches {AIj : j = 1, . . . , n}. So every finite subfamily of A = {AI0 : I0 ∈ Fin(I)} is patched. Since the sets FI0 , I0 ∈ Fin(I), are finite, we may apply the Patching Theorem to obtain a function F patching the whole A. Of course, F satisfies the desired condition. 2 A [partial] valuation on S is a [partial] function on S with values in {0, 1}. A family N of sets is a net iff K1 , K2 ∈ N implies K0 ⊆ K1 , K2 , for some K0 ∈ N . Theorem 6 (A. Robinson, see [1, Th. 4] Let Φ = {ϕt : t ∈ T } be a family of partial valuations on S and let N be a net of subsets of T . Let moreover, for every finite S0 ⊆ S and any K ∈ N there exists k ∈ K such that S0 ⊆ dm(ϕk ). Then there exists a valuation g, that for every finite S0 ⊆ S and every K ∈ N there exists k ∈ K with S0 ⊆ dm(ϕk ) and gdS0 = ϕk dS0 . Proof. Let J = N ×Fin(S) and let AK,S0 = S0 and FK,S0 = {ϕk dS0 : k ∈ K and S0 ⊆ dm(ϕk )}, for hK, S0 i ∈ J. Let J0 = {hK1 , S1 i, . . . , hKn , Sn i}, where K1 , . . . , Kn ∈ N and S1 , . . . , Sn ∈ Fin(S). Then there exists K0 ∈ N

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and k ∈ K0 such that K0 ⊆ K1 ∩ . . . ∩ Kn and S0 = S1 ∪ . . . ∪ Sn ⊆ dm(ϕk ). This, however, proves that F0 = ϕk patches {AhK0 ,S1 i , . . . , AhK0 ,Sn i }. Hence, by Patching Theorem, there exists F patching the whole {AhK,S0 i : K ∈ N , S0 ∈ Fin(S)}. Now, let S0 ∈ Fin(S) and let K ∈ N . Since F dS0 ∈ FK,S0 , there exists such k ∈ K that F dS0 = ϕk dS0 and S0 ⊆ dm(ϕk ), which proves that g = F satisfies the desired property. 2 Cowen in [2] proves the compactness of so called Finite Constraint Satisfaction. Before, we formulate it precisely, we have to present some definitions. A first order language is a tuple L = hF, P, V, τ i, where F, P and V are disjoint sets. The elements of F and P are called functional and predicate letters. Elements of V are called individual variables. Moreover τ is a function with domain P ∪ F with nonnegative integers as values such that τ (P) 6= 0, for any P ∈ P. Those of f ∈ F with τ (f) = 0 are constants of L. We define the set of terms of L as the least set TrmL) with V ⊆ Trm(L) and satisfying t1 , . . . , tn ∈ Trm(L), f ∈ F, τ (f) = n =⇒ ft1 . . . tn ∈ Im(L) An atomic formula is a string Pt1 . . . tn , where t1 , . . . , tn are terms and P is a predicate letter with τ (P) = n. Given a first order language L, we define a model of L as a tuple A =< A, I >, where A is any nonempty set and I is a function with domain P ∪ F such that I(P) ⊆ Aτ (P) and I(f) : Aτ (f) → A. A valuation in a model A is any mapping v : V → A. Every model A and a valuation v in A defines a function [v] which takes terms of L as arguments and yields an element of A as the value. This value is defined by induction of the length of the term as follows: [v](x) = v(x), if x ∈ V, and [v](ft1 . . . tn ) = I(f)([v](t1 ), . . . , [v](tn )). We say that an atomic formula Pt1 . . . tn is true in the model A under an assignment v iff < [v](t1 ), . . . , [v](tn ) >∈ I(P). A valuation satisfies a set X of atomic formulas iff it makes all formulas of X true. A set of atomic formulas is satisfiable in a model A iff there is a valuation in A which satisfies X. Now we can formulate Cowen’s Finite Constraint Satisfaction. Here we give it in a slightly different formulation (IN = {1, 2, 3, . . .} - the set of positive integers).

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Theorem 7. Let L be a first order language and let R = {Φj : j ∈ J} be a family of atomic formulas of L. Moreover, let A be a model of L and let us assume, that there exist such finite Dk ⊆ |A|, k ∈ IN, that, for every finite R0 ⊆ R there exists a valuation v0 in A satisfying R and such that v0 (xk ) ∈ Dk , k ∈ IN. Then there exists a valuation v in A satisfying R with v(xk ) ∈ Dk , k ∈ IN. Proof. Let Aj be the set of all individual variables of Φj and let Fj = {v : dm(v) = Aj , v - satisfies Φj and ∀xi ∈Aj (v(xi ) ∈ Di )}, j ∈ J. Then given a finite J0 ⊆ J, there exists an interpretation v0 satisfying {Φj : j ∈ J0 } and with v(xk ) ∈ Dk , k ∈ IN. Then v0 patches {Aj : j ∈ J0 }. Hence, by Patching Theorem, there exists a valuation v patching {Aj : j ∈ J}. Of course v satisfies R. 2 Let S be a set. A partition of S is a pairwise disjoint family of sets whose join is S. A property P of sets is of finite character, if a set belongs to P iff all its finite subsets belong to P. Let (P1 , . . . , Pn ) be properties and S a set. A (P1 , . . . , Pn )-partition of S is such a partition (S1 , . . . , Sn ) of S that Si is in Pi , i ∈ {1, . . . , n}. Cowen, Hechler and Mih´ok in [3], utilize the following compactness property of being (P1 , . . . , Pn )-partitionable: Theorem 8 (Set Partition Theorem). Let S be a set and let P1 , . . . , Pn be properties of finite character on S. Then there exists a (P1 , . . . , Pn )partition of S iff there exists a (P1 , . . . , Pn )-partition of every finite subset of S. Proof. Let J = Fin(S), and let AS0 = S0 and FS0 = {f : S0 → {1, . . . , n}|f −1 ”{j} ∈ Pj , j = 1, . . . , n}, for S0 ∈ Fin(S). Now, S let J0 ⊆ J be finite. To see that A0 = {Aj : j ∈ J0 } is patched, let S0 = J0 . Of course S0 is finite, hence there is a (P1 , . . . , Pn )-partition (S1 , . . . , Sn ) of S0 , and then F0 : S0 → {1, . . . , n}, defined as F0 (s) = i, for s ∈ Si , i = 1, . . . , n, patches A0 . Indeed, let S 0 ∈ J0 and i ∈ {1, . . . , n}. Then (F0 dS 0 )−1 ”{i} = (F0−1 ”{i}) ∩ S 0 = Si ∩ S 0 is in Pi , for it is a subset of Si and Pi is of finite character. Thus F0 dS 0 ∈ FS 0 , for each S 0 ∈ J0 . Hence, there exists F patching the entire A = {Aj : j ∈ J}. Let Si = F −1 ”{i}, i = 1, . . . , n. We will show that (S1 , . . . , Sn ) is a (P1 , . . . , Pn )-partition of S.

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To do this, we need to show that each Si is in Pi , i = 1, . . . , n. Since all Pi are of finite character, it suffices to show that all finite subsets of Si are in Pi , i = 1, . . . , n. So let i ∈ {1, . . . , n} and let S0 ⊆ Si = F −1 ”{i} be finite. Then, since F patches A, there is F dS0 ∈ FS0 . That is there exists f : S0 → {1, . . . , n} satisfying f −1 ”{j} ∈ Pj , j = 1, . . . , n and f = F dS0 . But then Pi 3 f −1 ”{i} = (F dS0 )−1 ”{i} = S0 ∩ (F −1 ”{i}) = S0 which completes the proof. 2

Remarks. It was stated that the Patching Theorem can be easily obtained in Zermelo– Fraenkel set theory, ZF, from Principle of Consistent Choices. It seems natural to ask whether this can be reversed. We have: Remark 1. Theorem.

The Patching Theorem implies the CT – the Compactness

Proof. Let X be a family of propositional sentences whose every finite subset is satisfiable. Using the Patching Theorem, we will show that X is satisfiable. Let J = X and let Aα be the set of all propositional variables in α and let Fα = {v : dm(v) = At(α), v – satisfying α}, α ∈ X. Then, if X0 is a finite subset of X, the family {Fα : α ∈ X0 } is patched by any v0 satisfying X0 . Hence, some v patches {Fα : α ∈ X}. This, however means that v satisfies X. As it is well known, this implies the desired result. 2 Since CT is equivalent to the Principle of Consistent Choices, the latter is equivalent to the Patching Theorem in ZF. Remark 2. The version of the Patching Theorem with infinite families of functions is not true. Proof. Indeed. Let J = IN, Aj = {0, 1, . . . , j}, j ∈ IN, and let Fj = (n) (n) (n) {fj : n = j + 1, j + 2, . . .}, where fj : Aj → IN is defined by fj (0) = n (n)

and fj (i) = i, i = 1, . . . , j. Then, given a number N ∈ IN, the function F0 : AN → IN, given by F0 (i) = i, i 6= 0, and F0 (0) = N patches the family {Aj : j = 0, . . . , N }. Let us suppose that there exists a function F patching

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the entire family {Aj : j ∈ IN}. Let M = F (0). Then, since F patches (n) {Aj : j ∈ IN}, we get F dAM ∈ FM , that is F dAM = fM , for some n > M . (n) Then, however, M = F (0) = fM (0) = n < M , which is a contradiction. This completes the proof. 2

References [1] R. H. Cowen, Some Combinatorial Theorems Equivalent to the Prime Ideal Theorem, Proc. Amer. Math. Soc. 41 (1973), pp. 268– 273. [2] R. H. Cowen, A Compactness Theorem for Infinite Constraint Satisfaction, Reports on Math. Logic 32 (1998), pp. 97–108. [3] R. H. Cowen, S. H. Hechler, P. Mih´ok, Graph Coloring Compactness Theorems Equivalent to BPI, Scientiae Mathematicae Japanicae 56.2 (2002), pp. 213–223. [4] P. E. Howard, Binary Consistent Choices on Pairs and a Generalisation of K¨ onig’s Infinity Lemma, Fund. Math. CXXI (1984), pp. 17–23. [5] A. Levy, Remarks on a Paper by J. Mycielski, Acta Mathematica XIV/1-2, pp. 126–130. [6] J. L Ã o´s, C. Ryll-Nardzewski, Application of Tychonoff ’s Theorem in Mathematical Proofs, Fund. Math. 38 (1951), pp. 233–237.

Institute of Computer Science Dep. of Mathematics and Information Technology Jagiellonian University Nawojki 11 30-072 Cracow Poland e-mail: [email protected]