Asymmetric regular types Slavko Moconja Joint work with Predrag - PowerPoint PPT Presentation

Asymmetric regular types Slavko Moconja Joint work with Predrag Tanovi´ c Faculty of Mathematics, Belgrade, Serbia Slavko Moconja (Belgrade) Asymmetric regular types 1 / 17

Invariant types Let p ( x ) ∈ S 1 (M) be a global type, and small A ⊂ M. Type p ( x ) is A − invariant if f ( p ) = p , for every f ∈ Aut A (M). Slavko Moconja (Belgrade) Asymmetric regular types 2 / 17

Invariant types Let p ( x ) ∈ S 1 (M) be a global type, and small A ⊂ M. Type p ( x ) is A − invariant if f ( p ) = p , for every f ∈ Aut A (M). Fact. If p ( x ) is A − invariant and B ⊇ A , then p ( x ) is B − invariant. Slavko Moconja (Belgrade) Asymmetric regular types 2 / 17

Regular types Let p ( x ) ∈ S 1 (M) be a global non-algebraic type and small A ⊂ M. Pair ( p ( x ) , A ) is regular if: 1 p ( x ) is A − invariant and 2 for every a � p | A and every small B ⊇ A : either a � p | B or p | B ⊢ p | Ba . Slavko Moconja (Belgrade) Asymmetric regular types 3 / 17

Regular types Let p ( x ) ∈ S 1 (M) be a global non-algebraic type and small A ⊂ M. Pair ( p ( x ) , A ) is regular if: 1 p ( x ) is A − invariant and 2 for every a � p | A and every small B ⊇ A : either a � p | B or p | B ⊢ p | Ba . Fact. If ( p ( x ) , A ) is a regular pair and B ⊇ A , then ( p ( x ) , B ) is a regular pair. Slavko Moconja (Belgrade) Asymmetric regular types 3 / 17

Asymmetric types Let p ( x ) ∈ S 1 (M) be a global non-algebraic A − invariant type. Type p ( x ) is asymmetric if for some B ⊇ A and Morley sequence ( a , b ) in p over B : ab �≡ ba ( B ). Slavko Moconja (Belgrade) Asymmetric regular types 4 / 17

Asymmetric types Let p ( x ) ∈ S 1 (M) be a global non-algebraic A − invariant type. Type p ( x ) is asymmetric if for some B ⊇ A and Morley sequence ( a , b ) in p over B : ab �≡ ba ( B ). Theorem Suppose that pair ( p ( x ) , A ) is regular and p ( x ) is asymmetric. Then there exists a finite extension A 0 of A and A 0 − definable partial order ≤ such that every Morley sequence in p over A 0 is strictly increasing. A. Pillay, P. Tanovi´ c, Generic stability, regularity and quasiminimality Slavko Moconja (Belgrade) Asymmetric regular types 4 / 17

cl p , A Let ( p ( x ) , A ) be a regular pair. Assume that p ( x ) is asymmetric over A . For X ⊆ ( p | A )(M) we define closure cl p , A ( X ) ⊆ ( p | A )(M) with: cl p , A ( X ) = { a � p | A | a � p | AX } . For small B ⊂ ( p | A )(M) we set: cl p , A , B ( X ) = cl p , A ( BX ) . Also, if M is some small model that contains A we define: cl M p , A ( X ) = cl p , A ( X ) ∩ M and cl M p , A , B ( X ) = cl p , A , B ( X ) ∩ M . Slavko Moconja (Belgrade) Asymmetric regular types 5 / 17

scl p , A For a � p | A we define symmetric closure scl p , A ( a ) ⊆ ( p | A )(M) with: scl p , A ( a ) = { b ∈ cl p , A ( a ) | a ∈ cl p , A ( b ) } . For X ⊆ ( p | A )(M) we define symmetric closure scl p , A ( X ) ⊆ ( p | A )(M) with: � scl p , A ( X ) = scl p , A ( a ) . a ∈ X We also define scl p , A , B , scl M p , A and scl M p , A , B . Slavko Moconja (Belgrade) Asymmetric regular types 6 / 17

Some facts about cl p , A and scl p , A 1 p | AX ⊢ p | A cl p , A ( X ); 2 cl p , A ( cl p , A , B ) is closure operator on ( p | A )(M); 3 cl p , A ( a 1 , a 2 , . . . , a n ) = cl p , A ( a ), where a is any maximal element in { a 1 , a 2 , . . . , a n } ; 4 ( a , b ) is Morley sequence in p over AB iff a / ∈ cl p , A ( B ) and b / ∈ cl p , A ( Ba ); � 5 cl p , A ( X ) = scl p , A ( a ); ( ∃ x ∈ X ) a ≤ x 6 ( p | A )(M) / scl p , A = { scl p , A ( a ) | a � p | A } is a partition of ( p | A )(M); 7 ( p | A )(M) / scl M p , A = { scl M p , A ( a ) | a � p | A } is a partition of ( p | A )(M) (M is small model that contains A ). Slavko Moconja (Belgrade) Asymmetric regular types 7 / 17

Order on ( p | A )(M) / scl p , A Lemma Suppose that scl p , A ( a ) � = scl p , A ( b ) and a < b. Then for every x ∈ scl p , A ( a ) and y ∈ scl p , A ( b ) is x < y. If scl p , A ( a ) � = scl p , A ( b ) and a � < b, then b < a. Corollary. Set ( p | A )(M) / scl p , A is linearly ordered. Slavko Moconja (Belgrade) Asymmetric regular types 8 / 17

Order on ( p | A )(M) / scl p , A Lemma Suppose that scl p , A ( a ) � = scl p , A ( b ) and a < b. Then for every x ∈ scl p , A ( a ) and y ∈ scl p , A ( b ) is x < y. If scl p , A ( a ) � = scl p , A ( b ) and a � < b, then b < a. Corollary. Set ( p | A )(M) / scl p , A is linearly ordered. Lemma Maximal Morley sequence in p over A in some small model M that contains A is exactly any set of representatives of ( p | A )(M) / scl M p , A partition. Corollary. Any two maximal Morley sequences in p over A in M have the same order-type. Slavko Moconja (Belgrade) Asymmetric regular types 8 / 17

Non-definable scl p , A Theorem Assume that scl p , A ( a ) is not Aa − definable, for some (every) a ∈ ( p | A )(M) . Then, for every countably order type there exists a countable model M such that the maximal Morley sequence in p over A in M has that order type. Corollary. If there exists global A − invariant, regular and asymmetric type whose scl p , A is not Aa − definable, then there are 2 ℵ 0 non-isomorphic countable models. Slavko Moconja (Belgrade) Asymmetric regular types 9 / 17

Example of asymmetric regular types Let M be a model of small o − minimal theory, p ∈ S 1 ( A ) non-algebraic type, and M monster model. Fact. p (M) is convex set. We have four kinds of p : (isolated type) there exist c , d ∈ dcl ( A ) such that c < x < d ⊢ p ( x ); (non-cut) there exist c ∈ dcl ( A ) and strictly decreasing sequence ( d n ) in dcl ( A ) such that { c < x < d n | n ∈ ω } ⊢ p ( x ); (non-cut) there exist strictly increasing sequence ( c n ) in dcl ( A ) and d ∈ dcl ( A ) such that { c n < x < d | n ∈ ω } ⊢ p ( x ); (cut) there exist strictly increasing sequence ( c n ) and strictly decreasing sequence ( d n ) in dcl ( A ) such that { c n < x < d n | n ∈ ω } ⊢ p ( x ). Slavko Moconja (Belgrade) Asymmetric regular types 10 / 17

Left and right global extensions: Case I Assume that there exists c ∈ dcl ( A ) such that c determines p ”on the left side”. Then for every M − formula φ , either φ or ¬ φ has interval that contains ( c , t ), for some t ∈ p (M). We define left global extension of p : p L ( x ) = { φ ( x ) | φ (M) contains ( c , t ), for some t ∈ p (M) } ∈ S 1 (M). Similarly we define right global extension p R of p , if there exists d ∈ dcl ( A ) such that d determines p ”on the right side”. Slavko Moconja (Belgrade) Asymmetric regular types 11 / 17

Left and right global extensions: Case II Assume that there exists strictly increasing sequence ( c n ) such that ( c n ) determines p ”on the left side”. Then for every M − formula φ , either φ or ¬ φ has interval that contains all but finitely many c n . We define left global extension of p : p L ( x ) = { φ ( x ) | φ (M) contains all but finitely many c n } ∈ S 1 (M). Similarly we define right global extension p R of p , if there exists strictly decreasing sequence ( d n ) such that ( d n ) determines p ”on the right side”. Slavko Moconja (Belgrade) Asymmetric regular types 12 / 17

p L and p R Theorem Both p L and p R are A − invariant, regular and asymmetric extensions of p. Moreover, p L and p R are the only two global A − invariant extensions of p. Any Morley sequence in p R is strictly increasing, and any Morley sequence in p L is strictly decreasing. Slavko Moconja (Belgrade) Asymmetric regular types 13 / 17

scl p L , A , scl p R , A Lemma Let a ∈ p (M) . Then: scl p L , A ( a ) = scl p R , A ( a ) = convex closure ( dcl ( Aa ) ∩ p (M)) . Corollary. I ⊂ p (M) is a Morley sequence in p L over A in M iff it is Morley sequence in p R over A in M. Also, I ⊆ p ( M ) is a maximal Morley sequence in p L over A in M iff it is maximal Morley sequence in p R over A in M, for any small model M that contains A . Remark. If p ∈ S 1 ( A ), then for some (any) a ∈ p (M), scl p L , A ( a ) is Aa − definable iff scl p L , A ( a ) = { a } . Slavko Moconja (Belgrade) Asymmetric regular types 14 / 17

⊥ w , �⊥ w , dimension Let p , q be two complete types (with parameters). We say that p ⊥ w q iff p ( x ) ∪ q ( y ) ⊢ tp ( xy ). �⊥ w is equivalence relation on S 1 ( ∅ ). Let { p i | i ∈ I } be the set of non-algebraic representatives of this equivalence relation. Slavko Moconja (Belgrade) Asymmetric regular types 15 / 17

⊥ w , �⊥ w , dimension Let p , q be two complete types (with parameters). We say that p ⊥ w q iff p ( x ) ∪ q ( y ) ⊢ tp ( xy ). �⊥ w is equivalence relation on S 1 ( ∅ ). Let { p i | i ∈ I } be the set of non-algebraic representatives of this equivalence relation. Let M be any countable model, A i = maximal Morley sequence in p iL , and � A = A i . i ∈ I Theorem M is prime over A. M and N are isomorphic iff maximal Morley sequence in p iL in M , and maximal Morley sequence in p iL in N have the same order-type, for every i ∈ I. Slavko Moconja (Belgrade) Asymmetric regular types 15 / 17