Reversible sequences of natural numbers and reversibility of some - - PowerPoint PPT Presentation

Reversible sequences of natural numbers and reversibility

• f some disconnected binary structures

Nenad Moraˇ ca (joint work with Miloˇ s S. Kurili´ c)

Department of Mathematics and Informatics, Faculty of Sciences, University of Novi Sad, Serbia

3rd July 2018

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Preliminaries

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Preliminaries

• A structure is called reversible iff all its bijective endomorphisms are

automorhpisms

(SETTOP 2018, Novi Sad) 3rd July 2018 2 / 13

Preliminaries

• A structure is called reversible iff all its bijective endomorphisms are

automorhpisms

• The class of reversible structures contains, for example, compact

Hausdorff and Euclidean topological spaces, linear orders, Boolean lattices, well founded posets with finite levels, tournaments, Henson graphs and Henson digraphs

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Preliminaries

• A structure is called reversible iff all its bijective endomorphisms are

automorhpisms

• The class of reversible structures contains, for example, compact

Hausdorff and Euclidean topological spaces, linear orders, Boolean lattices, well founded posets with finite levels, tournaments, Henson graphs and Henson digraphs

• extreme elements of L∞ω-definable classes of interpretations under some

syntactical restrictions are reversible (Kurili´ c, M.)

(SETTOP 2018, Novi Sad) 3rd July 2018 2 / 13

Preliminaries

• A structure is called reversible iff all its bijective endomorphisms are

automorhpisms

• The class of reversible structures contains, for example, compact

Hausdorff and Euclidean topological spaces, linear orders, Boolean lattices, well founded posets with finite levels, tournaments, Henson graphs and Henson digraphs

• extreme elements of L∞ω-definable classes of interpretations under some

syntactical restrictions are reversible (Kurili´ c, M.)

• monomorphic (or chainable) structures are reversible (Kurili´

c)

(SETTOP 2018, Novi Sad) 3rd July 2018 2 / 13

Preliminaries

• A structure is called reversible iff all its bijective endomorphisms are

automorhpisms

• The class of reversible structures contains, for example, compact

Hausdorff and Euclidean topological spaces, linear orders, Boolean lattices, well founded posets with finite levels, tournaments, Henson graphs and Henson digraphs

• extreme elements of L∞ω-definable classes of interpretations under some

syntactical restrictions are reversible (Kurili´ c, M.)

• monomorphic (or chainable) structures are reversible (Kurili´

c)

• the Rado graph, for example, is not reversible
• Reversible structures have the property Cantor-Schr¨
• der-Bernstein

(shorter CSB) for condensations (bijective homomorphisms)

(SETTOP 2018, Novi Sad) 3rd July 2018 2 / 13

Preliminaries

• A structure is called reversible iff all its bijective endomorphisms are

automorhpisms

• The class of reversible structures contains, for example, compact

Hausdorff and Euclidean topological spaces, linear orders, Boolean lattices, well founded posets with finite levels, tournaments, Henson graphs and Henson digraphs

• extreme elements of L∞ω-definable classes of interpretations under some

syntactical restrictions are reversible (Kurili´ c, M.)

• monomorphic (or chainable) structures are reversible (Kurili´

c)

• the Rado graph, for example, is not reversible
• Reversible structures have the property Cantor-Schr¨
• der-Bernstein

(shorter CSB) for condensations (bijective homomorphisms)

• each class of reversible posets yields the corresponding class of

reversible topological spaces

(SETTOP 2018, Novi Sad) 3rd July 2018 2 / 13

Preliminaries

• A structure is called reversible iff all its bijective endomorphisms are

automorhpisms

• The class of reversible structures contains, for example, compact

Hausdorff and Euclidean topological spaces, linear orders, Boolean lattices, well founded posets with finite levels, tournaments, Henson graphs and Henson digraphs

• extreme elements of L∞ω-definable classes of interpretations under some

syntactical restrictions are reversible (Kurili´ c, M.)

• monomorphic (or chainable) structures are reversible (Kurili´

c)

• the Rado graph, for example, is not reversible
• Reversible structures have the property Cantor-Schr¨
• der-Bernstein

(shorter CSB) for condensations (bijective homomorphisms)

• each class of reversible posets yields the corresponding class of

reversible topological spaces

• reversibility is related to the size of the classes [ρ]∼

=, and to the shape and

structure of certain suborders of the condensational order

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Disconnected binary structures

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Disconnected binary structures

In this presentation we investigate reversibility in the class of binary structures, that is models of the relational language Lb = R, where ar(R) = 2, and, moreover, we restrict our attention to the class of disconnected Lb-structures.

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Disconnected binary structures

In this presentation we investigate reversibility in the class of binary structures, that is models of the relational language Lb = R, where ar(R) = 2, and, moreover, we restrict our attention to the class of disconnected Lb-structures. If X = X, ρ is an Lb-structure and ∼ρ the minimal equivalence relation on X containing ρ, then the corresponding equivalence classes are called the connectivity components of X and X is said to be disconnected if it has more than one component, that is, if ∼ρ = X2). The prototypical disconnected structures are, of course, equivalence relations themselves; other prominent representatives of that class are some countable ultrahomogeneous graphs and posets, non-rooted trees, etc.

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Reversible sequences of cardinals

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Reversible sequences of cardinals

If X is a binary structure, and Xi, i ∈ I, are its connectivity components, then, clearly, the sequence of cardinal numbers |Xi| : i ∈ I is an isomorphism-invariant of the structure, and in some classes of structures (for example, in the class of equivalence relations) that cardinal invariant characterizes the structure up to isomorphism.

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Reversible sequences of cardinals

If X is a binary structure, and Xi, i ∈ I, are its connectivity components, then, clearly, the sequence of cardinal numbers |Xi| : i ∈ I is an isomorphism-invariant of the structure, and in some classes of structures (for example, in the class of equivalence relations) that cardinal invariant characterizes the structure up to isomorphism. In such classes the reversibility

• f a structure, being an isomorphism-invariant as well, can be regarded as a

property of the corresponding sequence of cardinals.

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Reversible sequences of cardinals

If X is a binary structure, and Xi, i ∈ I, are its connectivity components, then, clearly, the sequence of cardinal numbers |Xi| : i ∈ I is an isomorphism-invariant of the structure, and in some classes of structures (for example, in the class of equivalence relations) that cardinal invariant characterizes the structure up to isomorphism. In such classes the reversibility

• f a structure, being an isomorphism-invariant as well, can be regarded as a

property of the corresponding sequence of cardinals. So we isolate the following property of sequences of cardinals (called reversibility as well) which characterizes reversibility in the class of equivalence relations:

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Reversible sequences of cardinals

If X is a binary structure, and Xi, i ∈ I, are its connectivity components, then, clearly, the sequence of cardinal numbers |Xi| : i ∈ I is an isomorphism-invariant of the structure, and in some classes of structures (for example, in the class of equivalence relations) that cardinal invariant characterizes the structure up to isomorphism. In such classes the reversibility

• f a structure, being an isomorphism-invariant as well, can be regarded as a

property of the corresponding sequence of cardinals. So we isolate the following property of sequences of cardinals (called reversibility as well) which characterizes reversibility in the class of equivalence relations: a sequence of non-zero cardinals κi : i ∈ I is defined to be reversible iff ¬∃f ∈ Sur(I) \ Sym(I) ∀j ∈ I

i∈f −1[{j}] κi = κj,

where Sym(I) (resp. Sur(I)) denotes the set of all bijections (resp. surjections) f : I → I.

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Reversible sequences of natural numbers

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Reversible sequences of natural numbers

Next, we characterize reversible sequences of cardinals. First, we reduce the problem to characterizing the reversible sequences of natural numbers. Proposition A sequence of nonzero cardinals κi : i ∈ I is reversible iff it is a finite-one-sequence or a reversible sequence of natural numbers.

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Reversible sequences of natural numbers

Next, we characterize reversible sequences of cardinals. First, we reduce the problem to characterizing the reversible sequences of natural numbers. Proposition A sequence of nonzero cardinals κi : i ∈ I is reversible iff it is a finite-one-sequence or a reversible sequence of natural numbers. In order to give the characterization of reversible sequences of natural numbers, we first recall some definitions. If ni : i ∈ I ∈ IN, then I =

m∈N Im, where

Im := {i ∈ I : ni = m}.

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Reversible sequences of natural numbers

Next, we characterize reversible sequences of cardinals. First, we reduce the problem to characterizing the reversible sequences of natural numbers. Proposition A sequence of nonzero cardinals κi : i ∈ I is reversible iff it is a finite-one-sequence or a reversible sequence of natural numbers. In order to give the characterization of reversible sequences of natural numbers, we first recall some definitions. If ni : i ∈ I ∈ IN, then I =

m∈N Im, where

Im := {i ∈ I : ni = m}. A set K ⊆ N is called independent iff n ∈ K \ {n}, for all n ∈ K, where K \ {n} is the subsemigroup of the semigroup N, + generated by K \ {n}; by gcd(K) we denote the greatest common divisor of the numbers from K.

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Reversible sequences of natural numbers

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Reversible sequences of natural numbers

Theorem A sequence ni : i ∈ I ∈ IN is reversible iff either it is a finite-to-one sequence, or K = {m ∈ N : |Im| ≥ ω} is a nonempty independent set and gcd(K) divides at most finitely many elements of the set {ni : i ∈ I}.

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Reversible sequences of natural numbers

Theorem A sequence ni : i ∈ I ∈ IN is reversible iff either it is a finite-to-one sequence, or K = {m ∈ N : |Im| ≥ ω} is a nonempty independent set and gcd(K) divides at most finitely many elements of the set {ni : i ∈ I}. For example, if I is a nonempty set of arbitrary size, and ni : i ∈ I ∈ IN, then we have:

• if K = ∅ (which is possible if |I| ≤ ω), the sequence ni is reversible;

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Reversible sequences of natural numbers

Theorem A sequence ni : i ∈ I ∈ IN is reversible iff either it is a finite-to-one sequence, or K = {m ∈ N : |Im| ≥ ω} is a nonempty independent set and gcd(K) divides at most finitely many elements of the set {ni : i ∈ I}. For example, if I is a nonempty set of arbitrary size, and ni : i ∈ I ∈ IN, then we have:

• if K = ∅ (which is possible if |I| ≤ ω), the sequence ni is reversible;
• if K = {2, 5}, the sequence ni is reversible iff {ni : i ∈ I} is finite;

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Reversible sequences of natural numbers

Theorem A sequence ni : i ∈ I ∈ IN is reversible iff either it is a finite-to-one sequence, or K = {m ∈ N : |Im| ≥ ω} is a nonempty independent set and gcd(K) divides at most finitely many elements of the set {ni : i ∈ I}. For example, if I is a nonempty set of arbitrary size, and ni : i ∈ I ∈ IN, then we have:

• if K = ∅ (which is possible if |I| ≤ ω), the sequence ni is reversible;
• if K = {2, 5}, the sequence ni is reversible iff {ni : i ∈ I} is finite;
• if K = {4, 10}, then the sequence ni is reversible iff the set {ni : i ∈ I}

contains at most finitely many even numbers;

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Reversible sequences of natural numbers

Theorem A sequence ni : i ∈ I ∈ IN is reversible iff either it is a finite-to-one sequence, or K = {m ∈ N : |Im| ≥ ω} is a nonempty independent set and gcd(K) divides at most finitely many elements of the set {ni : i ∈ I}. For example, if I is a nonempty set of arbitrary size, and ni : i ∈ I ∈ IN, then we have:

• if K = ∅ (which is possible if |I| ≤ ω), the sequence ni is reversible;
• if K = {2, 5}, the sequence ni is reversible iff {ni : i ∈ I} is finite;
• if K = {4, 10}, then the sequence ni is reversible iff the set {ni : i ∈ I}

contains at most finitely many even numbers; Proposition (a) (NN)rev is a dense Fσδσ-subset of the Baire space NN;

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Reversible sequences of natural numbers

Theorem A sequence ni : i ∈ I ∈ IN is reversible iff either it is a finite-to-one sequence, or K = {m ∈ N : |Im| ≥ ω} is a nonempty independent set and gcd(K) divides at most finitely many elements of the set {ni : i ∈ I}. For example, if I is a nonempty set of arbitrary size, and ni : i ∈ I ∈ IN, then we have:

• if K = ∅ (which is possible if |I| ≤ ω), the sequence ni is reversible;
• if K = {2, 5}, the sequence ni is reversible iff {ni : i ∈ I} is finite;
• if K = {4, 10}, then the sequence ni is reversible iff the set {ni : i ∈ I}

contains at most finitely many even numbers; Proposition (a) (NN)rev is a dense Fσδσ-subset of the Baire space NN; (b) (NN)rev is not a subsemigroup of the semigroup NN, ◦.

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Reversible RFM structures

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Reversible RFM structures

We shalll say that a sequence of L-structures Xi : i ∈ I is rich for monomorphisms iff ∀i, j ∈ I ∀A ∈ [Xj]|Xi| ∃g ∈ Mono(Xi, Xj) g[Xi] = A. Since the reversibility of the components is a necessary condition for the reversibility of a disconnected binary structure, by RFM we denote the class

• f sequences Xi : i ∈ I (where I is any non-empty set) of pairwise disjoint,

connected and reversible Lb-structures, which are rich for monomorphisms.

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Reversible RFM structures

We shalll say that a sequence of L-structures Xi : i ∈ I is rich for monomorphisms iff ∀i, j ∈ I ∀A ∈ [Xj]|Xi| ∃g ∈ Mono(Xi, Xj) g[Xi] = A. Since the reversibility of the components is a necessary condition for the reversibility of a disconnected binary structure, by RFM we denote the class

• f sequences Xi : i ∈ I (where I is any non-empty set) of pairwise disjoint,

connected and reversible Lb-structures, which are rich for monomorphisms. Theorem If Xi : i ∈ I ∈ RFM then we have that

i∈I Xi is a reversible structure if and

• nly if |Xi| : i ∈ I is a reversible sequence of cardinals.

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Reversible RFM structures

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Reversible RFM structures

Theorem Let ∼ be an equivalence relation on a set X, X = X, ∼, and {Xi : i ∈ I} the corresponding partition. Then the structure X is reversible iff |Xi| : i ∈ I is a reversible sequence of cardinals. The same holds for the graphs (resp. posets) of the form X =

i∈I Xi, where

Xi, i ∈ I, are pairwise disjoint complete graphs (resp. cardinals ≤ ω).

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Reversible RFM structures

Theorem Let ∼ be an equivalence relation on a set X, X = X, ∼, and {Xi : i ∈ I} the corresponding partition. Then the structure X is reversible iff |Xi| : i ∈ I is a reversible sequence of cardinals. The same holds for the graphs (resp. posets) of the form X =

i∈I Xi, where

Xi, i ∈ I, are pairwise disjoint complete graphs (resp. cardinals ≤ ω). Remark There are c-many non-isomorphic countable reversible, as well as c-many non-isomorphic countable nonreversible equivalence relations. The same holds for the classes of graphs and posets from the previous theorem.

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Reversible countable ultrahomogeneous graphs

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Reversible countable ultrahomogeneous graphs

Using the above theorem, we can complete the characterization of reversible countable ultrahomogeneous graphs.

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Reversible countable ultrahomogeneous graphs

Using the above theorem, we can complete the characterization of reversible countable ultrahomogeneous graphs. By the well known characterization of Lachlan and Woodrow, each countable ultrahomogeneous graph is isomorphic to one of the following:

• Gµν - the union of µ disjoint copies of Kν, where µν = ω. Gµν is

reversible iff µ < ω or ν < ω;

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Reversible countable ultrahomogeneous graphs

Using the above theorem, we can complete the characterization of reversible countable ultrahomogeneous graphs. By the well known characterization of Lachlan and Woodrow, each countable ultrahomogeneous graph is isomorphic to one of the following:

• Gµν - the union of µ disjoint copies of Kν, where µν = ω. Gµν is

reversible iff µ < ω or ν < ω;

• GRado - the Rado graph. GRado is not reversible;

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Reversible countable ultrahomogeneous graphs

Using the above theorem, we can complete the characterization of reversible countable ultrahomogeneous graphs. By the well known characterization of Lachlan and Woodrow, each countable ultrahomogeneous graph is isomorphic to one of the following:

• Gµν - the union of µ disjoint copies of Kν, where µν = ω. Gµν is

reversible iff µ < ω or ν < ω;

• GRado - the Rado graph. GRado is not reversible;
• Hn - the Henson graph (for n ≥ 3). Hn is reversible since it is an extreme

structure, namely a maximal Kn free graph (Kurili´ c, M.);

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Reversible countable ultrahomogeneous graphs

Using the above theorem, we can complete the characterization of reversible countable ultrahomogeneous graphs. By the well known characterization of Lachlan and Woodrow, each countable ultrahomogeneous graph is isomorphic to one of the following:

• Gµν - the union of µ disjoint copies of Kν, where µν = ω. Gµν is

reversible iff µ < ω or ν < ω;

• GRado - the Rado graph. GRado is not reversible;
• Hn - the Henson graph (for n ≥ 3). Hn is reversible since it is an extreme

structure, namely a maximal Kn free graph (Kurili´ c, M.);

• Graph complements of these graphs. A graph is reversible iff its graph

complement is.

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More reversible digraphs, posets and topological spaces

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More reversible digraphs, posets and topological spaces

In the following theorem we detect a class of structures such that the reversibility of a structure belonging to the class follows from the reversibility

• f the sequence of cardinalities of its components. The converse is not true.

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More reversible digraphs, posets and topological spaces

In the following theorem we detect a class of structures such that the reversibility of a structure belonging to the class follows from the reversibility

• f the sequence of cardinalities of its components. The converse is not true.

Theorem If Xi, i ∈ I, are disjoint tournaments and the sequence of cardinals |Xi| : i ∈ I is reversible, then the digraph

i∈I Xi is reversible.

This statement holds if, in particular, Xi, i ∈ I, are disjoint linear orders. Then

• i∈I Xi is a reversible disconnected partial order.

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More reversible digraphs, posets and topological spaces

In the following theorem we detect a class of structures such that the reversibility of a structure belonging to the class follows from the reversibility

• f the sequence of cardinalities of its components. The converse is not true.

Theorem If Xi, i ∈ I, are disjoint tournaments and the sequence of cardinals |Xi| : i ∈ I is reversible, then the digraph

i∈I Xi is reversible.

This statement holds if, in particular, Xi, i ∈ I, are disjoint linear orders. Then

• i∈I Xi is a reversible disconnected partial order.

Let us recall that if P = P, ≤ is a partial order and O the topology on the set P generated by the base consisting of the sets of the form Bp := {q ∈ p : q ≤ p}, then endomorphisms of P are exactly the continuous self mappings of the space P, O.

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More reversible digraphs, posets and topological spaces

In the following theorem we detect a class of structures such that the reversibility of a structure belonging to the class follows from the reversibility

• f the sequence of cardinalities of its components. The converse is not true.

Theorem If Xi, i ∈ I, are disjoint tournaments and the sequence of cardinals |Xi| : i ∈ I is reversible, then the digraph

i∈I Xi is reversible.

This statement holds if, in particular, Xi, i ∈ I, are disjoint linear orders. Then

• i∈I Xi is a reversible disconnected partial order.

Let us recall that if P = P, ≤ is a partial order and O the topology on the set P generated by the base consisting of the sets of the form Bp := {q ∈ p : q ≤ p}, then endomorphisms of P are exactly the continuous self mappings of the space P, O. We conclude that the poset P is reversible iff P, O is a reversible topological space (i.e., each continuous bijection is a homeomorphism). So, previous theorem generates a large class of reversible topological spaces.

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Reversible disjoint unions of ordinals

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Reversible disjoint unions of ordinals

Lastly, using the characterization of reversible sequences of natural numbers

• btained above, we characterize reversible posets that are a disjoint union of
• rdinals αi = γi + ni, i ∈ I, where γi ∈ Lim ∪{0}, and ni ∈ ω.

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Reversible disjoint unions of ordinals

Lastly, using the characterization of reversible sequences of natural numbers

• btained above, we characterize reversible posets that are a disjoint union of
• rdinals αi = γi + ni, i ∈ I, where γi ∈ Lim ∪{0}, and ni ∈ ω.

Let us, before that, define Iα := {i ∈ I : αi = α}, for α ∈ Ord, Jγ := {j ∈ I : γj = γ}, for γ ∈ Lim ∪{0}.

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Reversible disjoint unions of ordinals

Lastly, using the characterization of reversible sequences of natural numbers

• btained above, we characterize reversible posets that are a disjoint union of
• rdinals αi = γi + ni, i ∈ I, where γi ∈ Lim ∪{0}, and ni ∈ ω.

Let us, before that, define Iα := {i ∈ I : αi = α}, for α ∈ Ord, Jγ := {j ∈ I : γj = γ}, for γ ∈ Lim ∪{0}. Theorem

• i∈I αi is a reversible poset iff exactly one of the following is true

(I) αi : i ∈ I is a finite-to-one sequence, (II) There is γ = max{γi : i ∈ I}, for α ≤ γ we have |Iα| < ω, and ni : i ∈ Jγ \ Iγ is a reversible sequence of natural numbers, which is not finite-to-one. The same holds for the poset

i∈I α∗ i .

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Reversible sequences of several things

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Reversible sequences of several things

Let us define a sequence of nonzero ordinals αi : i ∈ I to be a reversible sequence of ordinals iff it satisfies (I) or (II) from the previous theorem. Then we have:

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Reversible sequences of several things

Let us define a sequence of nonzero ordinals αi : i ∈ I to be a reversible sequence of ordinals iff it satisfies (I) or (II) from the previous theorem. Then we have: Proposition For each sequence of nonzero cardinals ¯ κ = κi : i ∈ I the following conditions are equivalent: (a) The poset

i∈I κi is a reversible structure;

(b) ¯ κ is a reversible sequence of ordinals; (c) ¯ κ is a reversible sequence of cardinals; (d) ¯ κ is a finite-to-one sequence or a reversible sequence of natural numbers.

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Reversible sequences of several things

Let us define a sequence of nonzero ordinals αi : i ∈ I to be a reversible sequence of ordinals iff it satisfies (I) or (II) from the previous theorem. Then we have: Proposition For each sequence of nonzero cardinals ¯ κ = κi : i ∈ I the following conditions are equivalent: (a) The poset

i∈I κi is a reversible structure;

(b) ¯ κ is a reversible sequence of ordinals; (c) ¯ κ is a reversible sequence of cardinals; (d) ¯ κ is a finite-to-one sequence or a reversible sequence of natural numbers. We remark that the equivalence (a) ⇔ (c) of the above proposition shows that the characterization of reversible RFM structures from a previous slide holds in a class of (sequences of) structures which is larger than the class RFM (for example, ω, ω1, ω2, ω3, . . . ∈ RFM).

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References

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c, N. Moraˇ ca, Reversible disjoint unions of well-orders and their inverses, Order (revised version submitted). https://arxiv.org/abs/1711.07053

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