Generically stable regular types
Abstract
We study nonorthogonality of symmetric, regular types and show that it preserves generic stability and is an equivalence relation on the set of all generically stable, regular types. We will also prove that some of the nice properties from the stable context hold in general. In the case of strongly regular types we will relate to the global RudinKeisler order.
The concept of (strong) regularity for global, invariant types in an arbitrary firstorder theory was introduced in Section 3 of [8]. The definition there was motivated by and extends that of regular (stationary) and strongly regular types in stable theories. Intuitively, it can be described as follows: Fix a global, invariant type . Consider all the formulas in as defining ”large” subsets of the monster and their negations as defining ”small” ones. Then it is natural to define: is the union of all small subsets definable over . It turned out that the regularity of means precisely that is a closure operation on the locus of (for almost all over which is invariant). There are two kinds of regular types:
 is symmetric. Morley sequences are totally indiscernible and the closure operation is a pregeometry operation inducing the dimension function.
 is asymmetric. The closure operation is induced by a definable partial ordering which totally orders Morley sequences.
Stable regular types are symmetric, while asymmetric regular types may exist only in theories with the strict order property. For example, the type of an infinite element in a theory of dense linear orders without endpoints is strongly regular. Interesting examples of both kinds of strongly regular types are heirs of ”generic” types of minimal and quasiminimal groups (and fields); they were recently studied in [6] and [2].
Asymmetric regular types are studied in detail in the forthcoming paper [7], and in this article we will concentrate on nonorthogonality of symmetric, regular types. For generically stable regular types we will prove that nonorthogonality is an equivalence relation.
Theorem 1.
Generic stability is preserved under nonorthogonality of regular types. Nonorthogonality is an equivalence relation on the set of all regular, generically stable types.
Next we will study generically stable, strongly regular types and prove that nonorthogonality is strongly related to the global version of Lascar’s RudinKeisler order which was originally defined in the stable context:
Theorem 2.
Suppose that is generically stable and strongly regular. Then:
(1) is RKminimal in the global RKorder.
(2) If is invariant then: if and only if .
The next theorem is probably more surprising than the previous, because its original proof in the stable case (see e.g [5]) relied heavily on the existence of prime models over arbitrary sets.
Theorem 3.
Suppose that and are invariant, strongly regular and generically stable. Then the following conditions are all equivalent:
(1) ;
(2) ;
(3) For some : .
The paper is organized as follows: Section 1 contains preliminaries. In Section 2 we will (slightly) redefine regularity for global invariant types. The redefinition is needed due to the fact that in Remark 3.1 in [8] it was noted (without proof) that the regularity condition in the definition does not depend on the particular choice of the parameter set over which the type is invariant. This is correct if the type is strongly regular but we do not know if that holds for an arbitrary regular type. Fortunately, the lapsus did not affect proofs of main results, only minor rephrasing of some of the statements is needed: replacement of ” is regular and invariant” by ” is regular over ”; the regularity of over a set is introduced in Definition 2.1 below. In Section 3 we study nonorthogonality of generically stable regular types and prove Theorem 1. The main technical fact used in the proof, stating that generically stable regular types have weight one, is proved in Proposition 3.3. Section 4 deals with strongly regular types and there we prove Theorems 2 and 3. As an application of the results from Section 3, in Section 5 we prove that one can vary dimensions of generically stable regular types in countable models as in the stable case:
Theorem 4.
Suppose that and are countable and is a countable family of pairwise orthogonal, regular over , generically stable types. Also assume that each is nonisolated. Then for any function there exists a countable such that for all .
1 Preliminaries
The notation is mainly standard, the only exception is the convention on the product of invariant types. We fix a complete firstorder theory and operate in its monster model . By we will denote elements and tuples of elements, by small subsets of the monster, while will denote small elementary submodels. Global types will be denoted by . A global type is invariant if whenever then for all with parameters from . is invariant if it is invariant over some small . will denote the restriction of to and is a Morley sequence in over if for all .
Assume for a while that is invariant. Then Morley sequences in over are indiscernible. We will occasionally go out of (into a larger monster) in order to get realizations of global types; these will be also denoted by , in which case will be welldefined due to the invariance of . Thus global Morley sequences are also welldefined, as well as the powers (types of Morley sequences of length ) are. Let . If then we will say that is symmetric; otherwise, it is asymmetric. If is symmetric, and is a permutation of then .
Products of invariant types were introduced in [3]. Here we will reverse the order in the definition: if and are invariant then their product is defined as follows: if and then ; thus our is the original . This change was suggested by Ludomir Newelski due to the fact that it is natural to have the equivalence: is a Morley sequence in over if and only if . The product is associative, but not commutative. We say that and commute if .
Complete types over the same domain are weakly orthogonal, or determines a complete type. Global types and are orthogonal, or , if they are weakly orthogonal. It is possible that both and invariant types (even regular in a superstable theory). The opposite situation, and may occur in an unstable theory, but not in a stable one. In a stable theory implies for all and, in particular, . We will see in Proposition 5.2 that this holds for any regular, generically stable type. For definable types over a model we have: hold for , if
Fact 1.1.
Suppose that and are both invariant and definable, and . Then for all ; in particular, .
A nonalgebraic global type is generically stable if, for some small , it is invariant and:
if is infinite and is a Morley sequence in over then for any formula (with parameters from ) is either finite or cofinite.
Using compactness it is straightforward to check that is generically stable if the condition holds for . Also, if is generically stable then as a witnessset in the definition we any small over which is invariant can be taken. Generically stable types are definable and symmetric. They commute with all invariant types. A power of a generically stable type may not be generically stable; an example the reader can find in [1]. However, this cannot happen if is in addition regular.
Let be any subset of the monster. A partial type is finitely satisfiable in if any finite subtype has infinitely many realization in , in which case we also say that is a type. By a sequence over we will mean a sequence such that is a type for each . Elements of a sequence can realize distinct types because there may exist many distinct types, so a sequence may not be indiscernible. The following well known fact guaranties existence of extensions.
Fact 1.2.
Suppose that a partial type is defined over and finitely satisfiable in . Then for any there exists a type in extending .
Nonisolation of can be expressed in terms of satisfiability: Fix and let . Then is nonisolated if and only if it is a type. Thus, isolation of a type is a strong negation of its finite satisfiability. The next fact follows from Fact 1.2.
Fact 1.3.
Suppose that is nonisolated, , and . Then has an extension in which is finitely satisfiable in .
A weak negation of satisfiability is the semiisolation: is semiisolated by over (or semiisolates over ), denoted also by , iff there is a formula such that ; is said to witness the semiisolation. Semiisolation is transitive: if is witnessed by and is witnessed by , then is witnessed by .
Fact 1.4.
Suppose that is nonalgebraic and . Let and assume . Then:
(1) witnesses if and only if it is not satisfied in .
(2) if and only if is a type.
The RudinKeisler order on complete types was introduced by Lascar in [4]. In an stable theory it was related to strong regularity; a nice exposition of the material can be found in Lascar’s book [5], or in Poizat’s book [9]. For nonalgebraic types define iff every model which realizes also realizes . is a quasiorder which particularly well behaves in an stable theory; there, due to the existence of prime models over arbitrary sets, omitting types is much easier than in general. In an stable theory RKminimal elements exist and they are precisely the strongly regular types. Some of equivalent ways of defining in the context are:

is realized in (the model prime over and a realization of );

There are and such that is isolated;

There are and such that .
In this article we will consider a variant of the RKorder, defined only for global types. In the unstable context only the third equivalent is adequate.
Definition 1.5.
if and only if there are and such that .
Transitivity of semiisolation implies that we have a quasiorder. However, the order can be quite trivial: take the theory of the random graph and notice that no two distinct global 1types are comparable.
If both and hold then we say that and are RKequivalent and denote it by . and are strongly RKequivalent, or , if there are and such that both and hold. RKequivalent types may not be strongly RKequivalent.
2 Regularity
In this section we will redefine regularity for global invariant types. To simplify notation we define it for global 1types only. This will not affect the generality because we can always switch to an appropriate sort in . The definition given here slightly differs in that we first define when is regular over (here is invariant), and then repeat the original one: is regular if such a small set exists. Concerning strong regularity, the definition remains unchanged.
Definition 2.1.
Let be a global nonalgebraic type and let be small.
(i) is said to be regular over if it is invariant and for any and : either or .
(ii) is regular if is regular over some small set.
Clearly, if is regular over and then is regular over , too. The same observation holds for strong regularity. But, before defining strong regularity it is convenient to introduce the following notation: we will say that is an invariant pair if is invariant and .
Definition 2.2.
(i) is strongly regular if for some small it is an invariant pair and:
for all and satisfying : either or .
(ii) is strongly regular if is strongly regular for some .
We will prove in Proposition 2.6 that as a witness set in the previous definition we can take any small for which is invariant. For, we need to label a local regularity condition.
Definition 2.3.
Suppose that and . We say that satisfies the weak orthogonality condition, or (WOR) for short, if:
WOR is a technical property of locally strongly regular types, see Definition 7.1 in [8]. Examples of such types are ”generic” types of minimal and quasiminimal structures. Recall that is a minimal structure iff any definable subset with parameters is either finite or cofinite. In a minimal structure there is a unique nonalgebraic type . satisfies WOR, so is locally strongly regular via . By Corollary 7.1 from [8] the same is true if is the ”generic” type of a quasiminimal structure (the type containing all the formulas with a cocountable solution set).
Remark 2.4.
Some of equivalent ways of expressing the fact that satisfies WOR are:

for all

Lemma 2.5.
Suppose that is invariant, , and satisfies WOR. Then satisfies WOR, too.
Proof.
Suppose, on the contrary, that is such that . Then there are realizing and a formula (over ) such that . Let be such that , and . Note that the invariance of implies that both and satisfy WOR. Since both and hold. In particular:
and
Let realize . Then . A contradiction. ∎
Proposition 2.6.
(i) An invariant pair is strongly regular if and only if satisfies WOR for all .
(ii) An invariant pair is strongly regular if and only if: for all and satisfying : either or . Therefore, as a witness set in the definition of strong regularity we can take any small set over which is invariant.
Proof.
(i) ) is easy, so we prove only ). Suppose that is strongly regular. Let be such that the regularity condition holds:
for all and satisfying : either or .
Fix and we will show that satisfies WOR (where ). Suppose that . Apply the regularity condition to : so ; then apply it to : so ; continuing in this way we get:
Thus and satisfies WOR. Now let . Then sastisfies WOR so, by Lemma 2.5, satisfies WOR, too.
(ii) Follows from part (i). ∎
Remark 2.7.
Suppose that is nonalgebraic and invariant. As in the proof of Proposition 2.6(i) one checks that is regular over if only if is invariant and satisfies WOR for any (finite) extension . The stronger equivalence, like the one that we have established for strongly regular types in Proposition 2.6, would be: an invariant type is regular iff is regular over . However that does not seem to hold: it is likely that there is a regular, invariant type which is not regular over (but we don’t know of an example).
The following fact, suggested by Anand Pillay, shows that the stronger equivalence holds for regular types under additional assumptions:
Proposition 2.8.
Suppose that is definable and invariant. Then is regular if and only if it is regular over .
Proof.
Only ) requires a proof. Suppose that is regular and let be such that is regular over . We claim that is regular over , too. Otherwise, for some there are realizing such that , and . Let be such that is a heir of . Then both and realize , because is a heir of .
Suppose and find and such that . Since is a coheir, there exists such that ; hence . A contradiction. We conclude .
Therefore , and , so is not regular over . A contradiction. ∎
Suppose that is invariant and let . Define as an operation on the power set of :
. for all
If is regular over then, by Lemma 3.1(iii) from [8], a closure operator on . The proof of this fact does not depend on Remark 3.1 there, neither does the proof of Theorem 3.1 there, which is a dichotomy theorem for regular types. Here we state only a restricted version and we will use only the first part:
Theorem 2.9.
Suppose that is regular over . Then is a closure operator on for all . We have two kinds of regular types:
(1) Symmetric ( is symmetric). Then is a pregeometry operator on for all .
(2) Asymmetric. Then there exists a finite extension of and an definable partial order such that every Morley sequence in over is strictly increasing; is not a pregeometry.
In this paper we will deal only with symmetric regular types. Then the pregeometry describes the independence: is a Morley sequence in over if and only if it is independent. In particular, maximal Morley sequences in any have the same cardinality, so is a well defined cardinal number.
3 Orthogonality
In this section we study orthogonality of regular symmetric types. Our goal is to prove Theorem 1. We start by mentioning a result from [7]; it will not be used further in the text:
Theorem 3.1.
A regular asymmetric type is orthogonal to any symmetric invariant type. In particular, symmetry is preserved under nonorthogonality of regular types.
Question 1.
Is an equivalence relation on the set of all regular symmetric types?
Below, a positive answer will be given for generically stable types.
Lemma 3.2.
Suppose that and are invariant and that is regular over and symmetric. Also suppose that and are such that does not realize and let . Then exactly one of the following two conditions holds:
(A) Whenever is a Morley sequence in over then
(B) is inconsistent.
Moreover, if is generically stable then (B) holds.
Proof.
Suppose that neither (A) nor (B) are satisfied and work for a contradiction. The failure of (A), by compactness, implies that for some
(1) 
We claim that is inconsistent. Otherwise it would be satisfied by some and whenever is a Morley sequence in over we would have ; this is justified by which is implied by: , the invariance of , and . Therefore realizes
which is in contradiction with (1).
Let be maximal such that is consistent and, without loss of generality, assume that realizes the type. The maximality of implies that no element of realizes , where denotes . Hence .
Fix and let be a Morley sequence in . For each choose such that ; note that holds. Let realize . is a Morley sequence in over for each , so the failure of (B) implies that is consistent; let realize it. Then implies . Since is symmetric, is a pregeometry so the independence of ’s implies that is a Morley sequence in . Thus realizes ; this contradicts the maximality of . ∎
In the next proposition we will prove that generically stable regular types ”have weight 1 with respect to the independence”.
Proposition 3.3.
Suppose that , and are (not necessarily distinct) invariant types and that is regular and generically stable. Let and . Then at least one of and holds.
Proof.
Suppose that neither of them holds and choose and . Let be . Let , . Then we have:
 and are invariant and is regular and symmetric;
 and ;
 does not realize and .
Corollary 3.4.
Suppose that , and are invariant and that is regular and generically stable. Then if and only if and .
Lemma 3.5.
Suppose that are invariant, and are regular over , and that is generically stable. Further, suppose that are realizations of (where ) respectively such that and . Then:
(i) .
(ii) For all witnessing and witnessing there exist and such that
Proof.
(i) Suppose on the contrary that . Let realize and let be a realization of such that . We claim that holds. Otherwise we have
which contradicts Proposition 3.3. The claim implies . Now choose such that . Then and , by regularity of , imply . Hence is a Morley sequence in over . By Proposition 3.3 at least one of and holds. The first is not possible because , and the second because . A contradiction.
(ii) By interpreting assumptions of the lemma and what we have just proved in part (i), we have:
By compactness there are , and such that:
Therefore and
This proves (ii). ∎
Proposition 3.6.
Generic stability is preserved under nonorthogonality of symmetric, regular types.
Proof.
Suppose that and are both regular, nonorthogonal and that is generically stable. Choose a small model such that both and are regular over and holds for their corresponding restrictions. Let , and be such that .
Suppose for a contradiction that is not generically stable. Then for a suitably chosen larger , over , and a Morley sequence in over there exists realizing
Since is symmetric, after possibly replacing the first and the second part of the sequence, we may assume . Also, after replacing the second part by a Morley sequence in over we may assume that each realizes .
For each choose such that . Then witnesses that . We claim that:
(2) 
To prove it note that