CCongruence of ultrafilters
Boris ˇSobot
Department of Mathematics and Informatics, University of Novi Sad,Trg Dositeja Obradovi´ca 4, 21000 Novi Sad, Serbiae-mail: [email protected]: 0000-0002-4848-0678
Abstract
We continue the research of the relation (cid:101) | on the set β N of ultrafilterson N , defined as an extension of the divisibility relation. It is a quasiorder,so we see it as an order on the set of = ∼ -equivalence classes, where F = ∼ G means that F and G are mutually (cid:101) | -divisible. Here we introduce a newtool: a relation of congruence modulo an ultrafilter. We first recall thecongruence of ultrafilters modulo an integer and show that = ∼ -equivalentultrafilters do not necessarily have the same residue modulo m ∈ N . Thenwe generalize this relation to congruence modulo an ultrafilter in a naturalway. After that, using iterated nonstandard extensions, we introduce astronger relation, which has nicer properties with respect to addition andmultiplication of ultrafilters. Finally, we also introduce a strengthening of (cid:101) | and show that it also behaves well in relation to the congruence relation. : 54D35, 54D80, 11A07, 11U10, 03H15 Keywords and phrases : divisibility, congruence, Stone- ˇCech compactifica-tion, ultrafilter, nonstandard integer
Let N be the set of natural numbers. The relation (cid:101) | , an extension of the divisi-bility relation | on N to the set β N of ultrafilters on N , was introduced in [12] andfurther investigated in [13–16]. The main idea was to understand the impact ofvarious properties of | to (cid:101) | and possibly, learning about the (cid:101) | -hierarchy, to ac-quire better understanding of | . In this paper we will make another step in thatdirection, considering possible extensions of the congruence relations to β N andtheir relation to (cid:101) | , as well as to the operations of addition and multiplicationon β N .When working with the set of ultrafilters βS on a set S it is common toidentify each element s ∈ S with the principal ultrafilter { A ⊆ S : s ∈ A } .Having that in mind, any binary operation (cid:63) on S can be extended to βS asfollows: for A ⊆ S , A ∈ F (cid:63) G ⇔ { s ∈ S : s − A ∈ G} ∈ F , (1)where s − A = { t ∈ S : s (cid:63) t ∈ A } . If ( S, (cid:63) ) is a semigroup equiped with thediscrete topology, ( βS, (cid:63) ) becomes a compact Hausdorff right-topological semi-group. The base sets for the topology are (clopen) sets A = {F ∈ βS : A ∈ F} .1 a r X i v : . [ m a t h . L O ] A ug any aspects of structures obtained in this way were examined in [7].Every function f : N → N can be extended uniquely to a continuous (cid:101) f : β N → β N : the ultrafilter (cid:101) f ( F ) is generated by { f [ A ] : A ∈ F} . This was used in[12] to define analogously an extension of a binary relation ρ on N to a relation (cid:101) ρ on β N : F (cid:101) ρ G if and only if for every A ∈ F the set ρ [ A ] := { n ∈ N : ( ∃ a ∈ A ) aρn } is in G . This coincides with the so-called canonical way of extending relationsfrom N to β N described in [6]. It turned out that the extension (cid:101) | of thedivisibility relation | has a simple equivalent definition, more convenient forpractical use: F (cid:101) | G ⇔ F ∩ U ⊆ G , where U = { A ∈ P ( N ) \ {∅} : A ↑ = A } is the family of sets upward closed for | . (cid:101) | is a quasiorder, so we think of it as an order on the set of = ∼ -equivalenceclasses, where F = ∼ G if and only if F (cid:101) | G and G (cid:101) | F . We say that C ⊆ N is convex if for all x, y ∈ C and all z such that x | z and z | y holds z ∈ C . Allultrafilters from the same = ∼ -equivalence class C have the same convex sets.Clearly, each equivalence class C is determined by F ∩ U (for any
F ∈ C ), or bythe family of convex sets belonging to any
F ∈ C .An ultrafilter F is divisible by some n ∈ N if and only if n N := { nk : k ∈ N } ∈ F . If F ∈ N as well, n (cid:101) | F holds if and only if n | F . Hence, we can writejust n | F in case n ∈ N .Especially useful are prime ultrafilters P : those (cid:101) | -divisible only by 1 andthemselves. This is equivalent to P ∈ P , where P is the set of prime numbers.The (cid:101) | -hierarchy can be naturally divided into two parts. The “lower”part, L , can be divided into levels: L = (cid:83) l<ω L l , where L l = { p p . . . p l : p , p , . . . , p l are prime } is the set of natural numbers having exactly l (not nec-essarily distinct) prime factors. Some nice properties of L were established in[14]; for example every ultrafilter in L l has exactly l prime ingredients (but be-ing divisible by the n -th power of a prime P is not the same as being divisible by P n times). The “upper” part, however, is much more complicated. It containsthe maximal = ∼ -class, M AX , consisting of ultrafilters divisible by all n ∈ N ,and consequently by all F ∈ β N ([15], Lemma 4.6). Another interesting class is N M AX , maximal among N - free ultrafiters (those that are not divisible by any n ∈ N ), see [16], Theorem 5.4. A set belonging to an N -free ultrafilter is calledan N - free set.The paper is organized as follows. In Section 2 several well-known results ofelementary number theory are employed to obtain results about the congruenceof ultrafilters modulo an integer in connection with the divisibility relation (cid:101) | .In Section 3 we recapitulate basic definitions about ω -hyperextensions, obtainedby iterating nonstandard extensions of the set Z . Tensor pairs play an importantrole here. They were first considered by Puritz in [11]; Di Nasso proved severaluseful characterizations and coined the term (see [4]). Most of the results in Sec-tion 3 are taken from Luperi Baglini’s thesis [8], where the concept of a tensor2air is implemented in the surrounding of ω -hyperextensions. In Section 4 wedefine congruence modulo an ultrafilter and find several conditions equivalentto this definition. The next section deals with a stronger relation, and we provesome results connecting it to (cid:101) | and operations of addition and multiplicationof ultrafilters. In Section 6 we define another version of divisibility, obtained ina natural way from the strong congruence relation, and get some basic resultsabout it. The last section contains several remarks and open questions. Notation. N is the set of natural numbers (without zero), ω = N ∪ { } , P is the set of prime numbers and Z the set of integers. The calligraphic letters F , G , H , . . . are reserved for ultrafilters, and small letters x, y, z, . . . for integers(both standard and nonstandard). For A ⊆ N , A ↑ = { n ∈ N : ∃ a ∈ A a | n } and A ↓ = { n ∈ N : ∃ a ∈ A n | a } . If m, r ∈ N , then Z m = { , , . . . , m − } and mA + r = { mn + r : n ∈ A } . Finally, U = { A ∈ P ( N ) \ {∅} : A ↑ = A } and V = { A ∈ P ( N ) \ { N } : A ↓ = A } .Because we use ∗ N for a nonstandard extension of N , to avoid confusion wewill not denote β N \ N with N ∗ . Likewise, we will avoid writing A for A × A ,since this notation had another meaning in papers preceding this one. Let m ∈ N and let Z m be given the discrete topology. The homomorphism h m : N → Z m is defined as follows: h m ( n ) is the residue of n modulo m . h m extends uniquely to a continuous function (cid:102) h m : β N → Z m . The next resultsfollows directly from [7], Corollary 4.22. Proposition 2.1 h m is a homomorphism, both for addition and multiplicationof ultrafilters. As described in the Introduction, the relation ≡ m of congruence modulo m can be extended to a relation (cid:103) ≡ m on β N : F (cid:103) ≡ m G if and only if, for every A ∈ F , { n ∈ N : ( ∃ a ∈ A ) n ≡ m a } ∈ G . Recall that the kernel of a function h : N → N is the relation ker h = { ( x, y ) ∈ N × N : h ( x ) = h ( y ) } . Proposition 2.2 ([12], Theorem 2.13) If h : N → N and ρ = ker h , then (cid:101) ρ = ker (cid:101) h . Thus, for m ∈ N the extension of ≡ m to β N coincides with the definitionfound in [7]: F (cid:103) ≡ m G if and only if h m ( F ) = h m ( G ). In particular, r < m is theresidue of F ∈ β N modulo m ( F (cid:103) ≡ m r ) if and only if m N + r ∈ F . For practicalreasons, we will denote the extension of ≡ m to β N also by ≡ m from now on.The congruence of ultrafilters modulo integer is not new, but it was mostlymarginally mentioned; for example the following interesting result has only thestatus of a comment in [7]. 3 roposition 2.3 ([7], Comment 11.20) For every F ∈ β N and every U ∈ F there is a neighborhood ¯ A of F such that A ⊆ U and for all G ∈ ¯ A \ A and all m ∈ N holds G ≡ m F . We begin with a simple result about the solvability of a system of congruencesin β N . A system such that its every finite subsystem has a solution in β N willbe called feasible . Lemma 2.4 (a) Let x ≡ m i a i (for i = 0 , , . . . , k , a i ∈ Z and m i ∈ N ) be afinite system of congruences. It has a solution in β N \ N if and only if it has asolution in N .(b) The system x ≡ m i a i (for i ∈ ω , a i ∈ Z and m i ∈ N ) of congruences hasa solution in β N if and only if it is feasible. Proof. (a) Let
F ∈ β N \ N be a solution of the given system. Then A i := { x ∈ N : x ≡ m i a i } ∈ F for each i = 0 , , . . . , k . Hence A := (cid:84) ki =0 A i ∈ F , and any x ∈ A is a solution of the given system.On the other hand, if s ∈ N is a solution and u = lcm ( m , m , . . . , m k ) (theleast common multiplier of m , m , . . . , m k ), then all the elements of the set B = { x ∈ N : x ≡ u s } are also solutions. Thus every F ∈ B \ B is a solution ofthe system in β N \ N .(b) One direction is trivial, so assume the given system to be feasible. Let A i = { x ∈ N : x ≡ m i a i } . By the assumption, every finite subsystem ofthe given system has a solution, so the family { A i : i < ω } has the finiteintersection property. Since all the sets A i are closed, it follows that A = (cid:84) i<ω A i is nonempty, and any F ∈ A is a solution of the given system. (cid:50) Since = ∼ -equivalence classes within L are singletons ([14], Corollary 5.10),each class in L trivially contains ultrafilters congruent only to one residue mod-ulo m . We want to investigate for which systems of congruences there is a= ∼ -class such that all its ultrafilters satisfy it. Clearly, such a system mustbe feasible. On the other hand, by Lemma 2.4 a feasible system S has asolution G ∈ β N so we can assume that it is a system of all congruencessatisfied by G (we will call such a system maximal ). Also, every congruence x ≡ m i r i is equivalent to a system of congruences modulo mutually prime fac-tors of m i , so we can assume that all m i are powers of primes themselves. Let Q S = { p ∈ P : G ≡ p n n ∈ N } and T S = P \ Q S . As a special case, if T S = ∅ , all ultrafilters from the class M AX satisfy S . A ⊂ N is an antichain if there are no distinct a, b ∈ A such that a | b . Theorem 2.5
For every maximal feasible system S of congruences x ≡ p n r p,n (for n ∈ ω , p ∈ P and r p,n < p n ) such that T S is infinite there is an = ∼ -equivalence class C (cid:54)⊆ L such that F ≡ p n r p,n for all F ∈ C . Proof.
We consider two cases. 4 ◦ Q is infinite. Let { q i : i ∈ ω } and { t i : i ∈ ω } be enumerations of Q S and T S respectively. For i ∈ ω let s i = min { n ∈ N : G (cid:54)≡ t ni } . We construct, byrecursion on n , a set A = { a n : n ∈ ω } such that a n < a n +1 and:(1) a n ∈ t s i + ni N + r t i ,s i + n for i < n ;(2) t s n n | a n ;(3) q nj | a n for every j < n .Start with choosing any a ∈ t s N . Assume that a n is constructed. We wantto choose a n +1 satisfying the system x ≡ t si + n +1 i r t i ,s i + n +1 for i ≤ n , x ≡ t sn +1 n +1 x ≡ q nj j ≤ n . By the Chinese remainder theorem this system has asolution in N such that a n +1 > a n . Clearly, obtained a n +1 satisfies conditions(1)-(3). A is an antichain: for all m < n , a m < a n implies that a n (cid:45) a m , and t s m m | a m and (1) imply that a m (cid:45) a n . Let C be the = ∼ -equivalence class of any ultrafil-ter containing A . Every ultrafilter F ∈ C contains A ↑ and A ↓ , so it contains A = A ↑ ∩ A ↓ as well. Condition (3) clearly implies that A intersects each level L l only in finitely many elements, so F / ∈ L , and in particular F is nonprincipal.By (1), A \ ( t s i + ni N + r t i ,s i + n ) is finite for all i and all n , hence F ≡ t si + ni r t i ,s i + n .By (3), F ≡ q ni i ∈ ω and n ∈ N . Thus F satisfies all congruences of thegiven system.2 ◦ Q is finite. We repeat the construction from case 1 ◦ , but for j ≥ | Q S | (when we “run out” of elements from Q S ) instead of q j in condition (3) we usesome elements t i ∈ T for i > n . (This condition is needed here only to ensurethat F / ∈ L .) (cid:50) Proposition 2.6 ([16], Lemma 5.2) If A is an N -free set, then A (cid:54)⊆ n N ∪ n N ∪ . . . ∪ n k N for any n , n , . . . , n k ∈ N \ { } . Example 2.7 (1) Let us show that the condition of T S being infinite in thetheorem above is necessary. Consider a system S consisting of x ≡ t i r i (forsome primes t , t , . . . , t l − and some nonzero r i < t i ) and x ≡ p n for all p ∈ P \ { t , t , . . . , t l − } and all n ∈ N . Let us show that there can be no = ∼ -class C such that all F ∈ C satisfy S . Assume the opposite. Then every such F contains all sets in U N := { A ∈ U : A is N -free } : by Proposition 2.6 each A ∈ U N must contain an element a mutually prime to all t , t , . . . , t l − . Hence a | F implies a N ∈ F , and therefore A ∈ F . This means that F ∩U = U N ∪{ n N : n ∈ N ∧ t i (cid:45) n for all i = 0 , , . . . , l − } . But now, if we change any of the r i ’sinto another nonzero value we stay inside the same class C .(2) In the class N M AX of (cid:101) | -maximal N -free ultrafilters one can find anultrafilter congruent to r modulo m for any < r < m such that gcd ( m, r ) = 1 .Namely, the family U N ∪ { N \ n N : n > } ∪ { m N + r } has the finite intersectionproperty: for any given A ∈ U N and n , n , . . . , n k ∈ N \ { } , since A is N -free,Proposition 2.6 says that there is a ∈ A mutually prime to all of m, n , . . . , n k .By the Chinese remainder theorem the system x ≡ m r , x (cid:54)≡ n i , x ≡ a has asolution, and it belongs to A ∩ ( m N + r ) ∩ (cid:84) ≤ i ≤ k ( N \ n i N ) . ∼ -class; first we need the following definition. A set S of residues modulo p ∈ P is a geometric set of residues if there are s and r such that 0 ≤ s < p , 0 < r < p and S = { rest ( sr k , p ) : k ∈ N } , where rest ( x, p )is the residue of x modulo p . Theorem 2.8
Let p ∈ P and let S ⊆ { , , . . . , p − } . There is an = ∼ -equivalence class C such that the set of residues of ultrafilters F ∈ C is exactly S if and only if S is a geometric set of residues. Proof. ( ⇐ ) First assume that S = { s , . . . , s l − } is a geometric set of residues,where s i = rest ( s r i , p ) (for i = 0 , , . . . , l −
1) are exactly all distinct residuesof numbers s r k modulo p . If S = { } , which happens for s = 0, any = ∼ -classof ultrafilters divisible by p (i.e. containg the set p N ) will do. Otherwise, byDirichlet’s prime number theorem, there are primes s ≡ p s and b ≡ p r . Let B = { sb k : k ∈ ω } , U (cid:48) = { U ∈ U : U ∩ B (cid:54) = ∅} and V (cid:48) = { V ∈ V : N \ V / ∈ U (cid:48) } .Then the family U (cid:48)(cid:48) = U (cid:48) ∪ V (cid:48) has the finite intersection property: U (cid:48) is closedfor finite intersections, and every V ∈ V (cid:48) contains B . Let C be the = ∼ -equivalence class determined by U (cid:48)(cid:48) . For every F ∈ C we have B ∈ F (since B ∪ { b k : k ∈ ω } ∈ V (cid:48) and N \ { b k : k ∈ ω } ∈ U (cid:48) ) and B ⊆ (cid:83) l − i =0 ( p N + s i ), soevery such F is congruent to some s i modulo p . On the other hand, for each i ∈ { , , . . . , l − } the family U (cid:48)(cid:48) ∪{ p N + s i } has the finite intersection property: B contains infinitely many elements from each of the sets p N + s i , and finiteintersections of sets from U (cid:48)(cid:48) contain all but finitely many elements from B , sothey also intersect p N + s i . Hence there is an ultrafilter F ∈ C such that
F ≡ p s i .( ⇒ ) Now assume S is the set of residues modulo p of ultrafilters F ∈ C for some = ∼ -equivalence class C . Every singleton is clearly a geometric set ofresidues (obtained by choosing the quotient r = 1), so we will assume | S | > W be the family of all convex sets belonging to all F ∈ C . Since theelements of S are all possible residues of ultrafilters F ∈ C , there is C ∈ W (a finite intersection of sets from ( U ∪ V ) ∩ F ) such that C ⊆ (cid:83) l − k =0 ( p N + s k )(otherwise W ∪{ N \ (cid:83) l − k =0 ( p N + s k ) } would have the finite intersection property).Let q be a primitive root modulo p (this means that for every 0 < r < p thereis k ∈ N such that q k ≡ p r ; see [3] for more details). Let S = { s , . . . , s l − } ,where s i = rest ( q k i , p ), k < k < . . . < k l − and for each s i the smallest k i ischosen. If we denote r i = k i − k for 0 < i < l , then s i = rest ( s q r i , p ).Claim 1. The set R := { r i : 0 < i < l } is closed for the gcd (greatest commondivisor) operation. Proof of Claim 1.
Let 0 < i < j < l . Take A to be the set of | -minimalelements of C ∩ ( p N + s ). By recursion on k , let A k +1 be the set of | -minimalelements of C ∩ A k ↑ ∩ ( p N + s i ), A k +2 the set of | -minimal elements of C ∩ A k +1 ↑ ∩ ( p N + s j ) and A k +3 the set of | -minimal elements of C ∩ A k +2 ↑∩ ( p N + s ). Each of the sets A m (for m ∈ ω ) must be nonempty, since otherwise C ⊆ ( C \ A ↑ ) ∪ ( C ∩ A ↑ \ A ↑ ) ∪ . . . ∪ ( C ∩ A m − ↑ ), and each of the (convex)6ets on the right would miss one of the sets p N + s , p N + s i or p N + s j , so itcould not belong to all ultrafilters in C .Now let d = gcd ( r i , r j ). By B´ezout’s lemma there are a (cid:48) , b (cid:48) ∈ Z such that a (cid:48) r i + b (cid:48) r j = d . By replacing a (cid:48) , b (cid:48) with their residues modulo p − a, b ∈ Z p − such that ar i + br j ≡ p − d . Let m = 3( a + b ) and let (cid:104) c i : 0 ≤ i < m (cid:105) be a | -chain in C of length m such that c i ∈ A i (it exists since A m − (cid:54) = ∅ ). Let c i +1 = c i d i ; then d k ≡ p q r i and d k d k +1 ≡ p q r j for all k . Hence e := d d . . . d a − d a d a +1 d a +1) d a +1)+1 . . . d a + b − d a + b − ≡ p ( q r i ) a ( q r j ) b = q ar i + br j ≡ p q d (in the last equality we used Fermat’s little theorem). But c e is divisible by c and divides c m ; since C is convex, c e ∈ C and hence d ∈ R .Claim 2. rest ( tr , p − ∈ R for all t ∈ N . Proof of Claim 2 is similar to (though simpler than) the one from Claim 1.We construct a | -chain (cid:104) c i : 0 ≤ i ≤ t − (cid:105) such that c i ∈ p N + s for odd i and c i ∈ p N + s for even i . If c i +1 = c i d i , we get c d d . . . d t − ≡ p q tr , so tr ≡ p − r j for some r j ∈ R .Now, since r < r < . . . < r l − , the two Claims show that R must have theform R = { ir : 0 < i < l } . But then s i ≡ s ( q r ) i , which is what we wanted toprove. (cid:50) ω -hyperextensions of Z In the previous two papers, [15] and [16], we employed nonstandard methods(more precisely, the superstructure approach) to get more information on therelation (cid:101) | . We will continue that practice here. However, now we turn toextensions of the set Z of all integers instead of N . The reason is, of course, thatwe want to use the operation of subtraction. Let X be a set containing a copyof Z consisting of atoms: none of the elements of X contains as an element anyof the other relevant sets. Let V ( X ) = X , V n +1 ( X ) = V n ( X ) ∪ P ( V n ( X )) for n ∈ ω and V ( X ) = (cid:83) n<ω V n ( X ). V ( X ) is then called a superstructure . Therank of an element x ∈ V ( X ) is the smallest n ∈ ω such that x ∈ V n ( X ).If V ( X ) is a superstructure, its nonstandard extension is a pair ( V ( Y ) , ∗ ),where V ( Y ) is a superstructure with the set of atoms Y and ∗ : V ( X ) → V ( Y )is a rank-preserving function such that A ⊆ ∗ A for A ⊆ X , Z ⊂ ∗ Z , ∗ X = Y and satisfying the Transfer principle (we delay the formulation of Transfer untillater, since we will need a more general version).A nonstandard extension ( V ( Y ) , ∗ ) of V ( X ) is a κ - enlargement if for everyfamily F of subsets of some set in V ( X ) with the finite intersection propertysuch that | F | < κ there is an element in (cid:84) A ∈ F ∗ A . κ -enlargements are knownto exist in ZFC. 7or an excellent introduction to nonstandard methods we refer the reader to[5]. The connection between a nonstandard extension and β Z is given by thefunction v : ∗ Z → β Z , defined by v ( x ) = { A ⊆ Z : x ∈ ∗ A } . v is onto whenever( V ( Y ) , ∗ ) is a c + -enlargement. Proposition 3.1 ([9], Lemma 1) For every x ∈ ∗ Z and every f : Z → Z , v ( ∗ f ( x )) = (cid:101) f ( v ( x )) . More information about v can be found in [9] and [8]. The following propo-sition is Theorem 3.1 of [15], adjusted for extensions of Z (instead of N ). Proposition 3.2
The following conditions are equivalent for every two ultra-filters F , G ∈ β Z :(i) F (cid:101) | G ;(ii) in every c + -enlargement V ( Y ) , there are x, y ∈ ∗ Z such that v ( x ) = F , v ( y ) = G and x ∗ | y ;(iii) in some c + -enlargement V ( Y ) , there are x, y ∈ ∗ Z such that v ( x ) = F , v ( y ) = G and x ∗ | y . First, let us establish that we can use all previously obtained results about ∗ N while working with ∗ Z . In every extension V ( Y ) the nonstandard set ∗ Z consists of ∗ N , another (“inverted”) copy of ∗ N (containing negative nonstandardnumbers) and zero. For x, y ∈ ∗ Z , x ∗ | y holds if and only if | x | ∗ | | y | .The situation with β Z is similar. Let, for A ⊆ Z , − A := {− a : a ∈ A } ;likewise, for F ∈ β N let −F := {− A : A ∈ F} . Then every ultrafilter in β Z (except the principal ultrafilter identified with zero) contains either N or − N , so β Z = β N ∪ {−F : F ∈ β N } ∪ { } . The family U Z := { U ∈ P ( Z ) \ {∅} : U ↑ = U } of upward closed subsets of Z consists of sets V ∪ − V ∪ { } for V ∈ U , anddivisibility in β Z is naturally defined as: F (cid:101) | G if and only if F ∩ U Z ⊆ G . Thus, F (cid:101) | G if and only if |F| (cid:101) | |G| (for absolute values of ultrafilters defined in theobvious way).We will write F − G instead of F + ( −G ). So A ∈ F − G if and only if { n ∈ Z : n − A ∈ G} ∈ F , where n − A = { n − a : a ∈ A } . Note that therecan be no confusion with this notation, since F − G is exactly the ultrafilterobtained by extending the subtraction operation from Z to β Z , as defined in(1).A nonstandard extension ( V ( Y ) , ∗ ) of V ( X ) is called a single superstructuremodel if Y = X . The existence of such model was proved in [1]. In a singlesuperstructure model it is possible to iterate the star-function, since it is definedfor all elements in the range of ∗ . Definition 3.3
Let ( V ( X ) , ∗ ) be a single superstructure model with Z ⊆ X .Define recursively, for x ∈ V ( X ) , S ( x ) = x and S k +1 ( x ) = ∗ ( S k ( x )) for all k ∈ ω . For A ⊆ X the set • A = (cid:83) k<ω S k ( A ) is called an ω -hyperextension of A . V ( X ) , S k ) is a nonstandard extension, and ( V ( X ) , • ) is also anonstandard extension. Moreover, we have the following. Proposition 3.4 ([8], Proposition 2.5.7) If ( V ( X ) , ∗ ) is a single superstructuremodel which is a c + -enlargement, then ( V ( X ) , S k ) for every k ∈ ω and ( V ( X ) , • ) are also c + -enlargements. We will call a single superstructure model ( V ( X ) , ∗ ) which is a c + -enlargementa ω - hyperenlargement .Now we can use the Transfer principle within any of the mentioned exten-sions. Recall that a first-order formula ϕ ( x , x , . . . , x n ) is bounded if all itsquantifiers are bounded, i.e. of the form ( ∀ x ∈ y ) or ( ∃ x ∈ y ). In the Transferprinciple the free variables x , x , . . . , x n that appear in ϕ ( x , x , . . . , x n ) cantake values of elements a , a , . . . , a n ∈ V ( X ) and in ϕ ( ∗ a , ∗ a , . . . , ∗ a n ) they arereplaced with their star-counterparts. Any k -ary relation A ∈ V ( X ) appearingas an atomic subformula in ϕ is also considered like a free variable and getsreplaced with ∗ A . The Transfer principle.
For every bounded formula ϕ and every a , a , . . . , a n ∈ V ( X ), in V ( X ) ϕ ( a , a , . . . , a n ) holds if and only if ϕ ( S k ( a ) , S k ( a ) , . . . , S k ( a n )) holds (for any k ∈ N ) if and only if ϕ ( • a , • a , . . . , • a n ) holds.As a simple application of Transfer let us show that ∗ ( x + y ) = ∗ x + ∗ y for x, y ∈ • Z , a fact that we will need later. If z = x + y , Transfer implies that ∗ z = ∗ x + ∗ y . Likewise, ∗ ( x · y ) = ∗ x · ∗ y . Proposition 3.5 ([8], Proposition 2.5.3)(a) For k ≤ l and A ⊆ Z , S k ( A ) = S l ( A ) ∩ S k ( Z ) . Consequently, S k ( A ) = • A ∩ S k ( Z ) .(b) For h : Z → Z and x ∈ S k ( Z ) , • h ( x ) = S k ( h )( x ) . Let us comment on the iterated version of the divisibility relation. It iscommon to omit ∗ (or, more generally, S k ) in formulas in front of the relations= and ∈ and arithmetical operations, in order to simplify notation. Let us showthat it is justified to do the same with the divisibility relation, even when workingin an ω -hyperextension. Firstly, ( x, y ) ∈ S k ( | ) can hold only if x, y ∈ S k ( Z ).On the other hand, for x ∈ S k ( N ), y ∈ S k ( Z ) and l > k , we will show that( x, y ) ∈ S k ( | ) if and only if ( x, y ) ∈ S l ( | ).( x, y ) ∈ S k ( | ) means that there is z ∈ S k ( Z ) such that y = xz . But S k ( Z ) ⊆ S l ( Z ), so ( x, y ) ∈ S l ( | ) follows. In the other direction, if ( x, y ) ∈ S l ( | ) for some l > k , and y = xz , then z ∈ S k ( Z ) so ( x, y ) ∈ S k ( | ) as well. Thus, there will beno ambiguity if we drop the stars and write simply x | y instead of ( x, y ) ∈ S k ( | ). Definition 3.6
For
F ∈ β Z , µ n ( F ) = { x ∈ S n ( Z ) : ( ∀ A ∈ F ) x ∈ S n ( A ) } .The monad of F is µ ( F ) = (cid:83) n<ω µ n ( F ) = { x ∈ • Z : ( ∀ A ∈ F ) x ∈ • A } .For x ∈ • Z , v ( x ) is the unique F ∈ β Z such that x ∈ µ ( F ) . Note that this definition of v ( x ) agrees with the previous one (for x ∈ ∗ Z ).9 roposition 3.7 ([8], Proposition 2.5.11) For every x ∈ • Z and every n ∈ ω , v ( S n ( x )) = v ( x ) . Let us recall the tensor (or Fubini) product of ultrafilters: for F , G ∈ β Z , F ⊗ G is the ultrafilter on Z × Z defined by S ∈ F ⊗ G ⇔ { x ∈ Z : { y ∈ Z : ( x, y ) ∈ S } ∈ G} ∈ F . The definitions of monads of ultrafilters of the form
F ⊗G and the correspondingfunction v are analogous as above. For ultrafilters F and G and nonstandardnumbers x ∈ µ ( F ) and y ∈ µ ( G ), ( x, y ) is a tensor pair if ( x, y ) ∈ µ ( F ⊗ G ). Lemma 3.8 If ( x, y ) ∈ ∗ Z × ∗ Z is a tensor pair, then so are ( x, − y ) and ( − x, y ) . Proof.
Let F = v ( x ) and G = v ( y ); then v ( − y ) = −G and v (( x, y )) = F ⊗ G .We need to prove that v (( x, − y )) = F ⊗ ( −G ). But whenever ( x, − y ) ∈ ∗ S forsome S ⊆ Z × Z , we have ( x, y ) ∈ ∗ S (cid:48) , where S (cid:48) := { ( m, − n ) : ( m, n ) ∈ S } . Bythe assumptions S (cid:48) ∈ F ⊗ G , so { x ∈ Z : { y ∈ Z : ( x, y ) ∈ S } ∈ ( −G ) } = { x ∈ Z : −{ y ∈ Z : ( x, y ) ∈ S } ∈ G} = { x ∈ Z : { y ∈ Z : ( x, y ) ∈ S (cid:48) } ∈ G} ∈ F , so S ∈ F ⊗ ( −G ).The proof for ( − x, y ) is analogous. (cid:50) By [4], Proposition 11.7.2, for any tensor pair ( x, y ) we have x + y ∈ µ ( F + G )and x · y ∈ µ ( F · G ). An important feature of ω -hyperextensions is that theyprovide a canonical way to obtain tensor pairs. Proposition 3.9 ([8], Theorem 2.5.27) If x ∈ µ ( F ) and y ∈ µ ( G ) , then thepair ( x, ∗ y ) is a tensor pair. Hence, x + ∗ y ∈ µ ( F + G ) and x · ∗ y ∈ µ ( F · G ) . A natural way to define the congruence relation modulo an ultrafilter would beto imitate again the construction of an extension (cid:101) ρ , as described in Section 2. Definition 4.1
For
M ∈ β N and F , G ∈ β Z , F ≡ M G if and only if for every A ∈ M the set { ( x, y ) ∈ Z × Z : ( ∃ m ∈ A ) x ≡ m y } belongs to the ultrafilter F ⊗ G . This definition has a nice equivalent formulation via divisibility of ultrafilters.
Lemma 4.2
For
M ∈ β N and F , G ∈ β Z , F ≡ M G if and only if M (cid:101) | F − G . Proof.
F ≡ M G ⇔ ( ∀ A ∈ M ) { x ∈ Z : { y ∈ Z : ( ∃ m ∈ A ) x ≡ m y } ∈ G} ∈ F⇔ ( ∀ A ∈ M ) { x ∈ Z : { y ∈ Z : x − y ∈ A ↑} ∈ G} ∈ F⇔ ( ∀ A ∈ M ∩ U ) { x ∈ Z : { y ∈ Z : x − y ∈ A } ∈ G} ∈ F⇔ ( ∀ A ∈ M ∩ U ) { x ∈ Z : x − A ∈ G} ∈ F⇔ ( ∀ A ∈ M ∩ U ) A ∈ F − G , M (cid:101) | F − G . (cid:50) The following lemma justifies our using the same notation as for the relationfrom Section 2.
Lemma 4.3 If m ∈ N and F , G ∈ β Z , F ≡ m G as defined in Section 2 isequivalent to F ≡ m G from Definition 4.1. Proof.
Since h m is a homomorphism, (cid:102) h m ( F − G ) = (cid:102) h m ( F ) − (cid:102) h m ( G ). It followsthat m | F − G if and only if (cid:102) h m ( F − G ) = 0, if and only if (cid:102) h m ( F ) − (cid:102) h m ( G ). (cid:50) ≡ M also has a nonstandard characterization. First we recall Puritz’s resultthat ( x, y ) ∈ ∗ N × ∗ N is a tensor pair if and only if x < ∗ f ( y ) for every f : N → N such that ∗ f ( y ) ∈ ∗ N \ N ([11], Theorem 3.4). Taking into account Lemma 3.8,we get the following version of this result. Proposition 4.4 ( x, y ) ∈ ∗ Z × ∗ Z is a tensor pair if and only if | x | < | ∗ f ( y ) | for every f : Z → Z such that ∗ f ( y ) ∈ ∗ Z \ Z . If we denote G = v ( y ), the condition ∗ f ( y ) / ∈ Z is equivalent to f (cid:22) B notbeing constant for any B ∈ G . Let us call f : Z → Z non- G -constant in thatcase.Note that we are still working in any c + -enlargement (we do not need an ω -hyperextension), so µ ( F ) here actually means µ ( F ). Theorem 4.5
Let
M ∈ β N and F , G ∈ β Z . The following conditions areequivalent:(i) F ≡ M G ;(ii) in some c + -enlargement holds ( ∀ m ∈ µ ( M ))( ∃ x ∈ µ ( F ))( ∃ y ∈ µ ( G ))(( x, y ) is a tensor pair ∧ m | x − y ) (2) (iii) in every c + -enlargement holds (2). Proof. (ii) ⇒ (i) Let (2) hold in some c + -enlargement. If y ∈ µ ( G ) then − y ∈ µ ( −G ). Since for a tensor pair ( x, y ) we have, by Lemma 3.8, x − y = x + ( − y ) ∈ µ ( F − G ), the “if” part follows directly from Proposition 3.2.(i) ⇒ (iii) Assume M (cid:101) | F − G ; we work in arbitrary c + -enlargement. Wedefine, for A, B ⊆ Z , M ⊆ N and f : Z → Z : E A,B,M = { ( m, a, b ) ∈ N × Z × Z : a ∈ A ∧ b ∈ B ∧ m ∈ M ∧ m | a − b } F f = { ( m, a, b ) ∈ N × Z × Z : | a | < | f ( b ) |} . We prove that the family { E A,B,M : A ∈ F , B ∈ G , M ∈ M} ∪ { F f : f : Z → Z is non- G -constant } has the finite intersection property. { E A,B,M : A ∈ F , B ∈G , M ∈ M} is closed for finite intersections. So let A ∈ F , B ∈ G , M ∈ M andlet f , f , . . . , f k : Z → Z be non- G -constant. Since M ↑∈ M ∩ U , M (cid:101) | F − G M ↑∈ F − G . Hence { n ∈ Z : n − M ↑∈ G} ∈ F . Let a ∈ A ∩ { n ∈ Z : n − M ↑∈ G} . This means that B := B ∩ ( a − M ↑ ) ∈ G . Hence there is b ∈ B such that | f i ( b ) | > | a | for all i ≤ k (otherwise { b ∈ B : f i ( b ) = j } ∈ G for some i ≤ k and some − a ≤ j ≤ a , a contradiction with the assumption that f i is non- G -constant). Since b ∈ a − M ↑ , there is m ∈ M such that m | a − b ,so ( m, a, b ) ∈ E A,B,M ∩ F f ∩ F f ∩ . . . ∩ F f k .Now, since we are working with a c + -enlargement, there is( m, x, y ) ∈ (cid:92) A ∈F ,B ∈G ,M ∈M ∗ E A,B,M ∩ (cid:92) f non- G -constant ∗ F f . This means that m ∈ µ ( M ), x ∈ µ ( F ), y ∈ µ ( G ) and m | x − y . Also, for everynon- G -constant f : Z → Z , | ∗ f ( y ) | > | x | , so ( x, y ) is a tensor pair. (cid:50) Unfortunately, we do not even know whether ≡ M is an equivalence relationon β Z , which makes it unconvenient to work with. Therefore in the next sectionwe introduce a stronger relation with much nicer properties. To better explain the forthcoming definition of congruence, we begin with a fewsimple lemmas. Recall that
M AX is the class of ultrafilters (cid:101) | -divisible by allothers. Lemma 5.1
Let x, y ∈ • Z and v ( x ) = v ( y ) . Then m | x − y for all m ∈ N and x − y ∈ µ ( M AX ) . Proof.
For each m ∈ N , let h m be the function defined in Section 2. Then • h m ( x ) ∈ Z m for all x ∈ • Z . By Proposition 3.1, v ( • h m ( x )) = (cid:102) h m ( v ( x )) = (cid:102) h m ( v ( y )) = v ( • h m ( y )), so x and y have the same residue modulo m .Ultrafilters from M AX are those divisible by all m ∈ N . Hence µ ( M AX )consists exactly of nonstandard numbers divisible by all m ∈ N , so the secondstatement follows directly from the first. (cid:50) By Theorem 2.8, the assumption of Lemma 5.1 can not be relaxed to v ( x ) = ∼ v ( y ): there are = ∼ -equivalent ultrafilters giving different residues modulo some m ∈ N . Lemma 5.2
Let x, y ∈ • Z , v ( x ) = v ( y ) and m ∈ S k ( N ) . Then m | S k ( x ) − S k ( y ) . Proof.
By Lemma 5.1, ( ∀ m ∈ N ) m | x − y . By Transfer, ( ∀ m ∈ S k ( N )) m | S k ( x ) − S k ( y ). (cid:50) Thus, for every m ∈ S k ( N ), all the numbers from µ ( F ) ∩ S k [ • Z ] have the sameresidue modulo m . We will use this to establish a strengthening of congruencemodulo M ∈ β N . 12 efinition 5.3 Ultrafilters F , G ∈ β Z are strongly congruent modulo M ∈ β N if, in every ω -hyperenlargement, ( ∀ m ∈ µ ( M ))( ∃ x ∈ µ ( F ))( ∃ y ∈ µ ( G )) m | ∗ x − ∗ y. (3) We write
F ≡ s M G . We easily get the following equivalent condition.
Lemma 5.4
F ≡ s M G implies that in every ω -hyperenlargement ( ∀ m ∈ µ ( M ))( ∀ x ∈ µ ( F ))( ∀ y ∈ µ ( G )) m | ∗ x − ∗ y. Proof.
Let x ∈ µ ( F ) and y ∈ µ ( G ) be such that m | ∗ x − ∗ y , and let x ∈ µ ( F )and y ∈ µ ( G ) be arbitrary. By Lemma 5.2, m | ∗ x − ∗ x and m | ∗ y − ∗ y , so m | ∗ x − ∗ y as well. (cid:50) To avoid constant repetition, in each of the proofs in the rest of the paperit will be understood that we are working in an ω -hyperenlargement (a singlestructure extension which is a c + -enlargement).It will follow from Lemmas 6.5, 6.3 and 4.2 that F ≡ s M G implies F ≡ M G .For now we prove that ≡ sm for m ∈ N also coincides with the congruence relationmodulo integer (from Section 2). Lemma 5.5 If m ∈ N and F , G ∈ β Z , F ≡ sm G holds if and only if F ≡ m G . Proof.
The only element of µ ( m ) is m itself. Let x ∈ µ ( F ) and y ∈ µ ( G )be such that m | ∗ x − ∗ y ; then ∗ x and ∗ y have the same residue modulo m : • h m ( ∗ x ) = • h m ( ∗ y ). Then, by Propositions 3.1 and 3.7, (cid:102) h m ( F ) = v ( • h m ( ∗ x )) = v ( • h m ( ∗ y )) = (cid:102) h m ( G ), so F ≡ m G . The other implication is proved similarly,using Lemma 5.4. (cid:50) Lemma 5.6 ≡ s M is an equivalence relation on the set β Z . Proof.
Reflexivity and symmetry are obvious from the definition. So let
F ≡ s M G and G ≡ s M H . By Lemma 5.4, for any m ∈ µ ( M ), x ∈ µ ( F ), y ∈ µ ( G ) and z ∈ µ ( H ) holds m | ∗ x − ∗ y and m | ∗ y − ∗ z . Then m | ∗ x − ∗ z , so F ≡ s M H . (cid:50) Theorem 5.7
Let
M ∈ β N . ≡ s M is compatible with operations + and · in β Z :(a) F ≡ s M F and G ≡ s M G imply F + G ≡ s M F + G ;(b) F ≡ s M F and G ≡ s M G imply F · G ≡ s M F · G . Proof.
Let m ∈ µ ( M ), x ∈ µ ( F ), x ∈ µ ( F ), y ∈ µ ( G ) and y ∈ µ ( G ). It follows from Proposition 3.7 that ∗ y ∈ µ ( G ) and ∗ y ∈ µ ( G ). Bythe assumptions we have m | ∗ x − ∗ x and m | ∗∗ y − ∗∗ y .(a) By Proposition 3.9 x + ∗ y ∈ µ ( F + G ) and x + ∗ y ∈ µ ( F + G ).From the above conclusions follows m | ( ∗ x + ∗∗ y ) − ( ∗ x + ∗∗ y ), i.e. m | ( x + ∗ y ) − ∗ ( x + ∗ y ). Since we started with arbitrary m ∈ µ ( M ), thismeans that F + G ≡ s M F + G .(b) By Proposition 3.9 x · ∗ y ∈ µ ( F · G ) and x · ∗ y ∈ µ ( F · G ). Wehave m | ( ∗ x − ∗ x ) ∗∗ y and m | ∗ x ( ∗∗ y − ∗∗ y ). Hence m | ∗ x ∗∗ y − ∗ x ∗∗ y ,i.e. m | ∗ ( x ∗ y ) − ∗ ( x ∗ y ), so F · G ≡ s M F · G . (cid:50) The following simple result is a version of a well-known fact ([10], Corollary8.3).
Lemma 5.8 (a) Every
F ∈
M AX is strongly congruent to zero modulo anyultrafilter;(b) for every
F ∈ β Z \ Z , F − F ∈
M AX . Proof. (a) For any
F ∈
M AX and any x ∈ µ ( F ), ( ∀ m ∈ N ) m | x implies byTransfer ( ∀ m ∈ ∗ N ) m | ∗ x , which gives us F ≡ M M .(b) We will show that A ∈ F − F for all A ∈ U Z . Let m ∈ A be arbitrary.Then there is r ∈ Z m such that m Z + r ∈ F , so since m Z ⊆ − A , it followsthat n − A ∈ F for all n ∈ m Z + r . Thus m Z + r ⊆ { n ∈ Z : n − A ∈ F} , so { n ∈ Z : n − A ∈ F} ∈ F , which means that A ∈ F − F . (cid:50) Let us also note, regarding the lemma above, that F = ∼ G is not enough toconclude that F − G ∈
M AX . By Theorem 2.8 there are F , G ∈ β N and m ∈ N such that F = ∼ G but F (cid:54)≡ m G , say F ≡ m r and G ≡ m r for some r < m and r < m . From Proposition 2.1 we get F − G ≡ m r − r (cid:54) = 0, so m (cid:45) F − G . Definition 5.9
A family {F i : i ∈ I } of ultrafilters is a complete residue systemmodulo M ∈ β N if it contains exactly one element of every equivalence class ofstrong congruence modulo M . As an application of the above results, we have an ultrafilter version of awell-known theorem on complete residue systems in Z . Theorem 5.10 If {F i : i ∈ I } is a complete residue system modulo M ∈ β N then, for every G ∈ β N , {F i + G : i ∈ I } and {G + F i : i ∈ I } are completeresidue systems modulo M . Proof.
We need to show that in R = {F i + G : i ∈ I } no two ultrafilters arecongruent modulo M , and that each congruence class has a representative in R . First assume F i + G ≡ s M F j + G for some i, j ∈ I , i (cid:54) = j . By Theorem 5.7 F i + G − G ≡ s M F j + G − G . By Lemma 5.8 F i = F i + 0 ≡ s M F i + G − G ≡ s M F j + G − G ≡ s M F j , a contradiction.Now let H ∈ β N be arbitrary. There is i ∈ I such that F i ≡ s M H − G . UsingTheorem 5.7 and Lemma 5.8 again we get F i + G ≡ s M H − G + G ≡ s M H .The proof that {G + F i : i ∈ I } is a complete residue system modulo M ∈ β N is analogous. (cid:50) Strong divisibility
It is natural to ask: which ultrafilters are strongly congruent to zero modulosome
M ∈ β N ? Are those exactly the ultrafilters divisible by M ? For example,we saw in Lemma 5.8 that (cid:101) | -maximal ultrafilters are always strongly congruentto zero. In general, the above question leads us to the following definition. Definition 6.1
Let
M ∈ β N and F ∈ β Z . F is strongly divisible by M if, inevery ω -hyperenlargement, ( ∀ m ∈ µ ( M ))( ∃ x ∈ µ ( F )) m | ∗ x. We write
M | s F . In the same way as Lemma 5.4, we get a seemingly stronger condition.
Lemma 6.2
M | s F implies that in every ω -hyperenlargement ( ∀ m ∈ µ ( M ))( ∀ x ∈ µ ( F )) m | ∗ x. Proposition 3.2 easily implies the following.
Lemma 6.3
For all
M ∈ β N and F ∈ β Z , M | s F implies M (cid:101) | F . It is tempting to try to prove the reverse implication; unfortunately this isnot true, as we will now see.
Lemma 6.4 No N -free ultrafilter has any | s -divisors. Proof.
Assume the opposite, that an N -free ultrafilter F is | s -divisible by some G . Then G is also N -free, so for any x ∈ µ ( F ) holds ( ∀ m ∈ N ) m (cid:45) x . By Transfer( ∀ m ∈ ∗ N ) m (cid:45) ∗ x , a contradiction with G (cid:45) s F . (cid:50) Thus, this notion of divisibility is too strong to be our main divisibilityrelation, but it has some properties that are in good accordance with the strongcongruence relation and operations on β N .However, Lemma 6.4 also says that | s is not reflexive: N -free ultrafilters arenot divisible by themselves. It is, however, transitive: let F | s G and G | s H .Let x ∈ µ ( F ), y ∈ µ ( G ) and z ∈ µ ( H ) be arbitrary. Then x | ∗ y and y | ∗ z .Hence ∗ y | ∗∗ z , so x | ∗∗ z , which suffices for F | s H . Lemma 6.5
F ≡ s M G if and only if M | s F − G . Proof. ( ⇒ ) Let m ∈ µ ( M ) be arbitrary and let x ∈ µ ( F ) and y ∈ µ ( G )be such that m | ∗ x − ∗ y . By Proposition 3.7, v ( y ) = v ( ∗ y ) so, by Lemma 5.2, m | ∗ y − ∗∗ y . It follows that m | ∗ x − ∗∗ y , i.e. m | ∗ ( x − ∗ y ). On the other hand, since − y ∈ µ ( −G ), by Lemma 3.8 and Proposition 3.9, x − ∗ y = x + ∗ ( − y ) ∈ µ ( F −G ),so
M | s F − G .( ⇐ ) Let m ∈ µ ( M ), x ∈ µ ( F ) and y ∈ µ ( G ) be arbitrary. Then x − ∗ y ∈ µ ( F − G ) so, by Lemma 6.2, m | ∗ ( x − ∗ y ). By Lemma 5.2 again we have m | ∗ y − ∗∗ y , so m | ∗ x − ∗ y , meaning that F ≡ s M G . (cid:50) heorem 6.6 Let
M ∈ β N and F , G ∈ β Z .(a) M | s F and M | s G imply M | s F + G ;(b) M | s F implies M | s F · G ;(c)
M | s G implies M | s F · G . Proof.
Let m ∈ µ ( M ), x ∈ µ ( F ) and y ∈ µ ( G ).(a) By assumptions m | ∗ x and m | ∗∗ y . Hence m | ∗ ( x + ∗ y ), and therefore M | s F + G .(b) Now we have m | ∗ x , which suffices for m | ∗ x ∗∗ y i.e. m | ∗ ( x ∗ y ), so M | s F · G .(c) By Lemma 6.2
M | s G implies m | ∗∗ y , so again m | ∗ x ∗∗ y and M | s F · G . (cid:50) Let us remind ourselves of the definitions of other three divisibility relationsfrom [12]:
G | L F iff ( ∃H ∈ β N ) F = H · GG | R F iff ( ∃H ∈ β N ) F = G · HG | M F iff ( ∃H , H ∈ β N ) F = H · G · H . What is the place of | s (restricted to β N × β N ) among these relations? Like allthe others, its restriction to N × N is just the usual divisibility relation (Lemma5.5). We already saw that | s ⊂ (cid:101) | . We will show that this is the only inclusionthat can be established:First, why | L (cid:54)⊆| s ? Let P , Q ∈ β N \ N be (cid:101) | -prime and let F = P · Q . Then
Q | L F but, by Lemma 6.4, Q (cid:45) s F . Analogously we conclude that | R (cid:54)⊆| s .That | s ⊆| M does not hold either can be seen by considering maximal classesof these two orders. By [13], Theorem 4.1, the | M -maximal ultrafilters areexactly those in the smallest ideal K ( β N , · ). On the other hand, the class of | s -maximal ultrafilters is exactly M AX by Lemmas 5.8 and 6.3. But
M AX is aproper superset of K ( β N , · ); we postpone the detailed examination of this andother aspects of maximal ultrafilters until a projected sequel to this paper. Even after finding, in Section 4, several equivalent conditions for ≡ M , we werenot able to answer the following. 16 uestion 7.1 Is ≡ M an equivalence relation? Not being able to prove that it is presents a big drawback for using thisrelation, which seems to be the most natural extension of the congruence relationto β N .Some more properties of our relations could be proved if we worked with c + -saturated nonstandard extensions. This is a stronger condition than beinga c + -enlargement: ( V ( Y ) , ∗ ) is κ - saturated if every family F of internal sets in V ( Y ) with the finite intersection property such that | F | < κ has nonempty in-tersection. To Proposition 3.2 one can add two more equivalent conditions (see[16], Theorem 3.4):(iv) in every c + -saturated extension V ( Y ), for every x ∈ µ ( F ) there is y ∈ µ ( G ) such that x ∗ | y ;(v) in every c + -saturated extension V ( Y ), for every y ∈ µ ( G ) there is x ∈ µ ( F ) such that x ∗ | y .However, Proposition 3.4 does not hold for c + -saturation in place of c + -enlargement: see [8], page 74. So to use the equivalents (iv) and (v) we wouldhave to answer the following question. Question 7.2
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