Equimorphism invariants for scattered linear orderings. Antonio - PowerPoint PPT Presentation

Equimorphism invariants for scattered linear orderings. Antonio Montalb´ an. SouthEastern Logic Symposium April 2006 Antonio Montalb´ an. Equimorphism invariants for scattered linear orderings.

Scattered linear orderings A linear ordering (a.k.a. total ordering) is a structure L = ( L , � ), where � is a transitive, reflexive and antisymmetric binary relation where every two elements are comparable. We say that A embeds into B , if A is isomorphic to a subset of B . We write A � B . Def : L is scattered if it doesn’t contain a copy of Q . Theorem: [Hausdorff ’08] Let S be the smallest class of linear orderings such that 1 ∈ S ; if A , B ∈ S , then A + B ∈ S ; and if κ is a regular cardinal and {A γ : γ ∈ κ } ⊆ S , then � � γ ∈ κ A i ∈ S and γ ∈ κ ∗ A i ∈ S . Then, S is the class of scattered linear orderings. Antonio Montalb´ an. Equimorphism invariants for scattered linear orderings.

Hausdorff rank Definition: Given a l.o. L , we define another l.o. L ′ by identifying the elements of L which have finitely many elements in between. Then we define L 0 = L , L α +1 = ( L α ) ′ , and take direct limits when α is a limit ordinal. rk( L ), the Hausdorff rank of L , is the least α such that L α is finite. Examples: rk( N ) = rk( Z ) = 1, rk( Z + Z + Z + · · · ) = 2, rk( ω α ) = α , rk( Q ) = ∞ . Observation: 1 if A � B , then rk( A ) � rk( B ); 2 rk( A + B ) = max(rk( A ) , rk( B )); 3 rk( A · B ) = rk( A ) + rk( B ); 4 A is scattered ⇔ for some α , A α is finite ⇔ rk( A ) � = ∞ . Antonio Montalb´ an. Equimorphism invariants for scattered linear orderings.

Fra¨ ıss´ e’s Conjecture Theorem: [Fra¨ ıss´ e’s Conjecture ’48; Laver ’71] The scattered linear orderings form a Well-Quasi-Ordering with respect to embeddablity. (i.e., there are no infinite descending sequences and no infinite antichains. ) Moreover, Laver proved that the class of σ -scattered linear orderings (countable union of scattered linear orderings) is Better-quasi-ordered with respect to emebeddability. Question: What is the proof theoretic strength of Fra¨ ıss´ e’s Conjecture? Antonio Montalb´ an. Equimorphism invariants for scattered linear orderings.

The structure of the scattered linear orderings Definition: A scattered L is indecomposable if whenever L � A + B , either L � A or L � B . Example: ω ∗ and ω 3 are indecomposable, but Z is not. Theorem: [Laver ’71] Every scattered linear ordering can be written as a finite sum of indecomposable ones. Theorem: [Fra¨ ıs´ e’s Conjecture ’48; Laver ’71] Every indecomposable linear ordering can be written either as a κ -sum or as a κ ∗ -sum of indecomposable l.o.’s of smaller rank, for some regular cardinal κ . Antonio Montalb´ an. Equimorphism invariants for scattered linear orderings.

Linear orderings - Equimorphism types We say that A and B are equimorphic if A � B and B � A . We denote this by A ∼ B . All the properties mentioned so far are preserved under equimorphisms. (scattered, indecomposable, rank, κ -sums, products...) Notation: Let S be the class of equimorphism types of scattered linear orderings. Let H ⊂ S be the class of equimorphism types of indecomposable linear orderings. To each L ∈ S we will assign a finite object with ordinal labels, Inv ( L ), such that A ∼ B ⇔ Inv ( A ) = Inv ( B ) . Antonio Montalb´ an. Equimorphism invariants for scattered linear orderings.

Def: A 0 + ... + A n is a minimal decomposition of L if each A i is indecomposable and n is minimal possible. Theorem: [Jullien ’69] Every scattered linear has a unique minimal decomposition, up to equimorphism. To each A ∈ H we will assign an invariant T ( A ) which is a finite tree with labels in O n × { + , −} such that A ∼ B ⇔ T ( A ) = T ( B ) . Then, we will then define Inv ( L )= � T ( A 0 ) , ..., T ( A n ) � . Antonio Montalb´ an. Equimorphism invariants for scattered linear orderings.

The structure of the indecomposables. Definition: L is indecomposable to the right if whenever L = A + B , L � B . If this is the case we let ǫ L = +. L is indecomposable to the left if whenever L = A + B , L � A . If this is the case we let ǫ L = − . Theorem [Jullien 69] Every scattered indecomposable linear ordering is indecomposable either to the right or to the left. Definition: Given a countable ordinal α , let H α = {L ∈ H : rk( L ) < α } . Given L ∈ H , let I L = {A ∈ H : 1 + A + 1 ≺ L} . Definition: Note that I L ⊆ H rk( L ) and that I L is and ideal . Antonio Montalb´ an. Equimorphism invariants for scattered linear orderings.

Theorem For A , B ∈ H , A ∼ B ⇔ ǫ A = ǫ B and I A = I B . Idea of the proof: Let κ = cf (rk( A )) ∨ ω . Lemma: I A has a cofinal subset of size κ . Let {A ξ : ξ < κ } ⊆ I A be a set cofinal in I A , where each memeber appears κ many times in the sequence. � Lemma: A ∼ A ξ . ξ ∈ κ ǫ A Lemma: κ = cf ( I A ) ∨ ω . � So, we get that A ξ depends only on ǫ A and I A . ξ ∈ κ ǫ A Antonio Montalb´ an. Equimorphism invariants for scattered linear orderings.

Finite Invariants Key observation: For every ideal I ⊂ H α , let X α I be the set of minimal elements of H α � I . Since H is a WQO, X α ∀L ∈ H α ( L ∈ I ⇔ ∀A ∈ X α I is finite and I ( A � � L )) . Definition Given L ∈ H of rank α , we define a finite tree T ( L ): Let X α I L = {A 0 , ..., A k } and let T ( L ) = � ǫ L , α � ������ � ���� � � � � ... � � � � ... ... � T ( A 0 ) T ( A k ) Recall that ǫ L = + if L is indec. to the right and ǫ L = − otherwise, and that I L = {A ∈ H : 1 + A + 1 ≺ L} Observation: For A , B ∈ H , A ∼ B ⇔ T ( A ) = T ( B ). Antonio Montalb´ an. Equimorphism invariants for scattered linear orderings.

Comparison of invariants for H The key point is that for A , B ∈ H , A � B if and only if either τ ( A ) � τ ( B ) and I A ⊆ I B , or τ ( A ) � � τ ( B ) and A ∈ I B . where τ ( L ) = ( cf (rk( L ) ∨ ω ) ǫ L Definition For S = [ � α, ǫ S � ; S 0 , ..., S l − 1 ] and T = [ � β, ǫ T � ; T 0 , ..., T k − 1 ] we let S � T if, either α � β , τ ( S ) � τ ( T ) and ∀ i < k (rk( T i ) � α ∨ ∃ j < l ( S j � T i )), or α < β , τ ( S ) � � τ ( T ) and ∀ i < k ( T i � � S ). ..,where rk( T ) = β and τ ( T ) = cf ( β ) ǫ T . Proposition For A , B ∈ H , A � B if and only if T ( A ) � T ( B ) . Antonio Montalb´ an. Equimorphism invariants for scattered linear orderings.

Comparison of invariants for S Key point: If A = A 0 + ... + A l and B = B 0 + ... + B k then A � B ⇔ A 0 + ... + A i 1 − 1 � B 0 & · · · & A i k + ... + A l � B k , for some 0 = i 0 � ... � i k � i k +1 = l + 1.. Definition Now, given S = � S 0 , ..., S l � and T = � T 0 , ..., T k � we let S � T if   �  �  . � S i n , S i n +1 , ..., S i n +1 − 1 � � T n 0= i 0 � ... � i k � i k +1 = l +1 n � k Proposition Let A , B ∈ S . Then, Inv ( A ) � Inv ( B ) if and only if A � B . Antonio Montalb´ an. Equimorphism invariants for scattered linear orderings.

Recognizing invariants Def: Let T r = { T ( L ) : L ∈ H } and I n = { Inv ( L ) : L ∈ S } . We are interested in characterizing T r and I n . Obs: A 0 + ... + A n is a minimal decomposition of L ∈ S , iff for no i < n we have A i + A i +1 ∼ A i or A i + A i +1 ∼ A i +1 . Obs: For L ∈ H of rank α , we have I L ⊆ H α has elements of arbitrary large rank < α . I L has the same cofinality as α , if infinite. Antonio Montalb´ an. Equimorphism invariants for scattered linear orderings.

Recognizing I n , the invariants for S . Obs: A i + A i +1 ∼ A i iff A i is indec. to the left and A i +1 ∈ I A i . Proposition Let T = � T 0 , ..., T k � ∈ T r <ω . Then, T ∈ I n if and only if for no i < k we have that 1 either ǫ i = − and T i +1 ∈ I T i , 2 or ǫ i +1 = + and T i ∈ I T i +1 , where I T = I α T 0 ,..., T k − 1 . Antonio Montalb´ an. Equimorphism invariants for scattered linear orderings.

Recognizing T r , the invariants for H . Let T r α = { T ∈ T r : rk( T ) < α } . Suppose we already know how to recognize the elements of T r α . Proposition A tree T = [ � α, ǫ � ; T 0 , ..., T k − 1 ] with labels in O n × { + , −} belongs to T r if and only if 1 for each i, T i ∈ T r α ; 2 T 0 , .., T k − 1 are mutually � -incomparable; 3 for no i, τ ( T i ) ≺ τ ( T ) . 4 rk( I α T 0 ,..., T k − 1 ) = α ; 5 cf ( I α T 0 ,..., T k − 1 ) ∨ ω = cf ( α ) ∨ ω ; where I α T 0 ,..., T k − 1 = { S ∈ T r α : rk( S ) < α & ∀ i < k ( T i � � S ) } . Given and ideal I ⊂ T r , let rk( I ) = sup { rk( T ) + 1 : T ∈ I} . Antonio Montalb´ an. Equimorphism invariants for scattered linear orderings.

Minimal Ideals So, to be able to recognize the elements of I n we need to recognize the ideals I ⊆ H α of rank α . Laver proved that H is a better-quasi-ordered (BQO), a stronger notion than wqo. Remark: The set of ideals of a BQO is also a BQO. So, the ideals of H α form, in particular, a WQO. Hence, there exists a finite set of minimal ideals of H α of rank α . If we found them we could tell whether an ideal has rank α by comparing it with these finitely many ideals. Antonio Montalb´ an. Equimorphism invariants for scattered linear orderings.