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POSETS OF COPIES, EMBEDDING MONOIDS, AND INTERPRETABILITY OF RELATIONAL STRUCTURES Milo s Kurili c Department of Mathematics and Informatics, University of Novi Sad, Serbia August 21, 2014 (SETTOP 2014) August 21, 2014 1 / 19 Posets

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SLIDE 1

POSETS OF COPIES, EMBEDDING MONOIDS, AND INTERPRETABILITY OF RELATIONAL STRUCTURES

Miloˇ s Kurili´ c

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

August 21, 2014

(SETTOP 2014) August 21, 2014 1 / 19

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SLIDE 2

Posets of copies of structures

Contents

(SETTOP 2014) August 21, 2014 2 / 19

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SLIDE 3

Posets of copies of structures

Contents

  • The poset of copies of a structure

(SETTOP 2014) August 21, 2014 2 / 19

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SLIDE 4

Posets of copies of structures

Contents

  • The poset of copies of a structure
  • Posets of copies and embedding monoids (under construction)

(SETTOP 2014) August 21, 2014 2 / 19

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SLIDE 5

Posets of copies of structures

Contents

  • The poset of copies of a structure
  • Posets of copies and embedding monoids (under construction)
  • Posets of copies of bi-interpretable structures (under construction)

(SETTOP 2014) August 21, 2014 2 / 19

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SLIDE 6

Posets of copies of structures

Relational structures and complete Boolean algebras

(SETTOP 2014) August 21, 2014 3 / 19

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SLIDE 7

Posets of copies of structures

Relational structures and complete Boolean algebras

If X = X, ρi : i ∈ I is a relational structure, by P(X) we denote the set of domains of isomorphic substructures of X, that is

(SETTOP 2014) August 21, 2014 3 / 19

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SLIDE 8

Posets of copies of structures

Relational structures and complete Boolean algebras

If X = X, ρi : i ∈ I is a relational structure, by P(X) we denote the set of domains of isomorphic substructures of X, that is P(X) =

  • A ⊂ X :
  • A, ρi ∩ Ani : i ∈ I
  • ∼

= X

  • (SETTOP 2014)

August 21, 2014 3 / 19

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SLIDE 9

Posets of copies of structures

Relational structures and complete Boolean algebras

If X = X, ρi : i ∈ I is a relational structure, by P(X) we denote the set of domains of isomorphic substructures of X, that is P(X) =

  • A ⊂ X :
  • A, ρi ∩ Ani : i ∈ I
  • ∼

= X

  • To X we adjoin
  • the poset P(X), ⊂

(SETTOP 2014) August 21, 2014 3 / 19

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SLIDE 10

Posets of copies of structures

Relational structures and complete Boolean algebras

If X = X, ρi : i ∈ I is a relational structure, by P(X) we denote the set of domains of isomorphic substructures of X, that is P(X) =

  • A ⊂ X :
  • A, ρi ∩ Ani : i ∈ I
  • ∼

= X

  • To X we adjoin
  • the poset P(X), ⊂
  • its separative quotient sqP(X), ⊂

(SETTOP 2014) August 21, 2014 3 / 19

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SLIDE 11

Posets of copies of structures

Relational structures and complete Boolean algebras

If X = X, ρi : i ∈ I is a relational structure, by P(X) we denote the set of domains of isomorphic substructures of X, that is P(X) =

  • A ⊂ X :
  • A, ρi ∩ Ani : i ∈ I
  • ∼

= X

  • To X we adjoin
  • the poset P(X), ⊂
  • its separative quotient sqP(X), ⊂
  • its Boolean completion ro sqP(X), ⊂

(SETTOP 2014) August 21, 2014 3 / 19

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SLIDE 12

Posets of copies of structures

Relational structures and complete Boolean algebras

If X = X, ρi : i ∈ I is a relational structure, by P(X) we denote the set of domains of isomorphic substructures of X, that is P(X) =

  • A ⊂ X :
  • A, ρi ∩ Ani : i ∈ I
  • ∼

= X

  • To X we adjoin
  • the poset P(X), ⊂
  • its separative quotient sqP(X), ⊂
  • its Boolean completion ro sqP(X), ⊂

Theorem ([9]) ro sqP(X), ⊂ is a homogeneous c. B. a.

(SETTOP 2014) August 21, 2014 3 / 19

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SLIDE 13

Posets of copies of structures

❅ ❅

  • ❅

❅

  • ❅

❅

  • ❅

❅

  • ❅

❅

  • ❅

❅

  • ❅

❅

HCBA+ CBA+ ro+ SEP sq Mod POSET Π

− → ↓ ↓

(SETTOP 2014) August 21, 2014 4 / 19

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SLIDE 14

Posets of copies of structures

❅ ❅

  • ❅

❅

  • ❅

❅

  • ❅

❅

  • ❅

❅

  • ❅

❅

  • ❅

❅

HCBA+ CBA+ ro+ SEP sq Mod POSET Π

− → ↓ ↓

ω, < q (SETTOP 2014) August 21, 2014 4 / 19

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SLIDE 15

Posets of copies of structures

❅ ❅

  • ❅

❅

  • ❅

❅

  • ❅

❅

  • ❅

❅

  • ❅

❅

  • ❅

❅

HCBA+ CBA+ ro+ SEP sq Mod POSET Π

− → ↓ ↓

ω, < q

q [ω]ω, ⊂

(SETTOP 2014) August 21, 2014 4 / 19

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SLIDE 16

Posets of copies of structures

❅ ❅

  • ❅

❅

  • ❅

❅

  • ❅

❅

  • ❅

❅

  • ❅

❅

  • ❅

❅

HCBA+ CBA+ ro+ SEP sq Mod POSET Π

− → ↓ ↓

ω, < q

q [ω]ω, ⊂ q (P(ω)/ Fin)+

(SETTOP 2014) August 21, 2014 4 / 19

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SLIDE 17

Posets of copies of structures

❅ ❅

  • ❅

❅

  • ❅

❅

  • ❅

❅

  • ❅

❅

  • ❅

❅

  • ❅

❅

HCBA+ CBA+ ro+ SEP sq Mod POSET Π

− → ↓ ↓

ω, < q

q [ω]ω, ⊂ q (P(ω)/ Fin)+ q

(ro P(ω)/ Fin)+ (SETTOP 2014) August 21, 2014 4 / 19

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SLIDE 18

Posets of copies of structures

❅ ❅

  • ❅

❅

  • ❅

❅

  • ❅

❅

  • ❅

❅

  • ❅

❅

  • ❅

❅

HCBA+ CBA+ ro+ SEP sq Mod POSET Π

− → ↓ ↓

ω, < q

q [ω]ω, ⊂ q (P(ω)/ Fin)+ q

(ro P(ω)/ Fin)+

q

Gω (SETTOP 2014) August 21, 2014 4 / 19

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SLIDE 19

Posets of copies of structures

❅ ❅

  • ❅

❅

  • ❅

❅

  • ❅

❅

  • ❅

❅

  • ❅

❅

  • ❅

❅

HCBA+ CBA+ ro+ SEP sq Mod POSET Π

− → ↓ ↓

ω, < q

q [ω]ω, ⊂ q (P(ω)/ Fin)+ q

(ro P(ω)/ Fin)+

q

Gω

q ω∗

(SETTOP 2014) August 21, 2014 4 / 19

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SLIDE 20

Posets of copies of structures

❅ ❅

  • ❅

❅

  • ❅

❅

  • ❅

❅

  • ❅

❅

  • ❅

❅

  • ❅

❅

HCBA+ CBA+ ro+ SEP sq Mod POSET Π

− → ↓ ↓

ω, < q

q [ω]ω, ⊂ q (P(ω)/ Fin)+ q

(ro P(ω)/ Fin)+

q

Gω

q ω∗ q

1 (SETTOP 2014) August 21, 2014 4 / 19

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SLIDE 21

Posets of copies of structures

❅ ❅

  • ❅

❅

  • ❅

❅

  • ❅

❅

  • ❅

❅

  • ❅

❅

  • ❅

❅

HCBA+ CBA+ ro+ SEP sq Mod POSET Π

− → ↓ ↓

ω, < q

q [ω]ω, ⊂ q (P(ω)/ Fin)+ q

(ro P(ω)/ Fin)+

q

Gω

q ω∗ q

1

q

D<ω2 (SETTOP 2014) August 21, 2014 4 / 19

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SLIDE 22

Posets of copies of structures

❅ ❅

  • ❅

❅

  • ❅

❅

  • ❅

❅

  • ❅

❅

  • ❅

❅

  • ❅

❅

HCBA+ CBA+ ro+ SEP sq Mod POSET Π

− → ↓ ↓

ω, < q

q [ω]ω, ⊂ q (P(ω)/ Fin)+ q

(ro P(ω)/ Fin)+

q

Gω

q ω∗ q

1

q

D<ω2

q <ω2

(SETTOP 2014) August 21, 2014 4 / 19

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SLIDE 23

Posets of copies of structures

❅ ❅

  • ❅

❅

  • ❅

❅

  • ❅

❅

  • ❅

❅

  • ❅

❅

  • ❅

❅

HCBA+ CBA+ ro+ SEP sq Mod POSET Π

− → ↓ ↓

ω, < q

q [ω]ω, ⊂ q (P(ω)/ Fin)+ q

(ro P(ω)/ Fin)+

q

Gω

q ω∗ q

1

q

D<ω2

q <ω2 q

(Borel /M)+ (SETTOP 2014) August 21, 2014 4 / 19

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SLIDE 24

Posets of copies of structures

❅ ❅

  • ❅

❅

  • ❅

❅

  • ❅

❅

  • ❅

❅

  • ❅

❅

  • ❅

❅

HCBA+ CBA+ ro+ SEP sq Mod POSET Π

− → ↓ ↓

ω, < q

q [ω]ω, ⊂ q (P(ω)/ Fin)+ q

(ro P(ω)/ Fin)+

q

Gω

q ω∗ q

1

q

D<ω2

q <ω2 q

(Borel /M)+

q

GZ (SETTOP 2014) August 21, 2014 4 / 19

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SLIDE 25

Posets of copies of structures

❅ ❅

  • ❅

❅

  • ❅

❅

  • ❅

❅

  • ❅

❅

  • ❅

❅

  • ❅

❅

HCBA+ CBA+ ro+ SEP sq Mod POSET Π

− → ↓ ↓

ω, < q

q [ω]ω, ⊂ q (P(ω)/ Fin)+ q

(ro P(ω)/ Fin)+

q

Gω

q ω∗ q

1

q

D<ω2

q <ω2 q

(Borel /M)+

q

GZ (SETTOP 2014) August 21, 2014 4 / 19

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SLIDE 26

Posets of copies of structures

Countable binary structures [3]

♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣

a t o m i c a t o m l e s s 1 ℵ0 > ℵ0 σ - c l o s e d a t o m l e s s 1 ℵ0 c i n d i v i s i b l e d i v i s i b l e i d e a l n

  • t

i d e a l F i n t a l l [ω]ω n

  • w

h e r e d e n s e i n [ω]ω ro sqP(X), ⊂ i s 2 Borel /M ro P(ω)/ Fin i s o m o r p h i c t o u n d e r CH |P(X)| X IX P(X) sqP(X), ⊂ | sqP(X), ⊂| P(X), ⊂ A1 A2 A3 B2 B3 C3 C4 D3 D4 D5 (SETTOP 2014) August 21, 2014 5 / 19

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SLIDE 27

Posets of copies of structures

Countable binary structures [3]

♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣

a t o m i c a t o m l e s s 1 ℵ0 > ℵ0 σ - c l o s e d a t o m l e s s 1 ℵ0 c i n d i v i s i b l e d i v i s i b l e i d e a l n

  • t

i d e a l F i n t a l l [ω]ω n

  • w

h e r e d e n s e i n [ω]ω ro sqP(X), ⊂ i s 2 Borel /M ro P(ω)/ Fin i s o m o r p h i c t o u n d e r CH |P(X)| X IX P(X) sqP(X), ⊂ | sqP(X), ⊂| P(X), ⊂ A1 A2 A3 B2 B3 C3 C4 D3 D4 D5 GZ (SETTOP 2014) August 21, 2014 5 / 19

slide-28
SLIDE 28

Posets of copies of structures

Countable binary structures [3]

♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣

a t o m i c a t o m l e s s 1 ℵ0 > ℵ0 σ - c l o s e d a t o m l e s s 1 ℵ0 c i n d i v i s i b l e d i v i s i b l e i d e a l n

  • t

i d e a l F i n t a l l [ω]ω n

  • w

h e r e d e n s e i n [ω]ω ro sqP(X), ⊂ i s 2 Borel /M ro P(ω)/ Fin i s o m o r p h i c t o u n d e r CH |P(X)| X IX P(X) sqP(X), ⊂ | sqP(X), ⊂| P(X), ⊂ A1 A2 A3 B2 B3 C3 C4 D3 D4 D5 GZ Gω (SETTOP 2014) August 21, 2014 5 / 19

slide-29
SLIDE 29

Posets of copies of structures

Countable binary structures [3]

♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣

a t o m i c a t o m l e s s 1 ℵ0 > ℵ0 σ - c l o s e d a t o m l e s s 1 ℵ0 c i n d i v i s i b l e d i v i s i b l e i d e a l n

  • t

i d e a l F i n t a l l [ω]ω n

  • w

h e r e d e n s e i n [ω]ω ro sqP(X), ⊂ i s 2 Borel /M ro P(ω)/ Fin i s o m o r p h i c t o u n d e r CH |P(X)| X IX P(X) sqP(X), ⊂ | sqP(X), ⊂| P(X), ⊂ A1 A2 A3 B2 B3 C3 C4 D3 D4 D5 GZ Gω D<ω2 (SETTOP 2014) August 21, 2014 5 / 19

slide-30
SLIDE 30

Posets of copies of structures

Countable binary structures [3]

♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣

a t o m i c a t o m l e s s 1 ℵ0 > ℵ0 σ - c l o s e d a t o m l e s s 1 ℵ0 c i n d i v i s i b l e d i v i s i b l e i d e a l n

  • t

i d e a l F i n t a l l [ω]ω n

  • w

h e r e d e n s e i n [ω]ω ro sqP(X), ⊂ i s 2 Borel /M ro P(ω)/ Fin i s o m o r p h i c t o u n d e r CH |P(X)| X IX P(X) sqP(X), ⊂ | sqP(X), ⊂| P(X), ⊂ A1 A2 A3 B2 B3 C3 C4 D3 D4 D5 GZ Gω D<ω2 ω, < (SETTOP 2014) August 21, 2014 5 / 19

slide-31
SLIDE 31

Posets of copies of structures

Countable binary structures [3]

♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣

a t o m i c a t o m l e s s 1 ℵ0 > ℵ0 σ - c l o s e d a t o m l e s s 1 ℵ0 c i n d i v i s i b l e d i v i s i b l e i d e a l n

  • t

i d e a l F i n t a l l [ω]ω n

  • w

h e r e d e n s e i n [ω]ω ro sqP(X), ⊂ i s 2 Borel /M ro P(ω)/ Fin i s o m o r p h i c t o u n d e r CH |P(X)| X IX P(X) sqP(X), ⊂ | sqP(X), ⊂| P(X), ⊂ A1 A2 A3 B2 B3 C3 C4 D3 D4 D5 GZ Gω D<ω2 ω, < Q (SETTOP 2014) August 21, 2014 5 / 19

slide-32
SLIDE 32

Posets of copies of structures

Countable binary structures [3]

♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣

a t o m i c a t o m l e s s 1 ℵ0 > ℵ0 σ - c l o s e d a t o m l e s s 1 ℵ0 c i n d i v i s i b l e d i v i s i b l e i d e a l n

  • t

i d e a l F i n t a l l [ω]ω n

  • w

h e r e d e n s e i n [ω]ω ro sqP(X), ⊂ i s 2 Borel /M ro P(ω)/ Fin i s o m o r p h i c t o u n d e r CH |P(X)| X IX P(X) sqP(X), ⊂ | sqP(X), ⊂| P(X), ⊂ A1 A2 A3 B2 B3 C3 C4 D3 D4 D5 GZ Gω D<ω2 ω, < Q XCol(ω,c) (SETTOP 2014) August 21, 2014 5 / 19

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SLIDE 33

Posets of copies of structures

The hierarchy of similarities between relational structures

(SETTOP 2014) August 21, 2014 6 / 19

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SLIDE 34

Posets of copies of structures

The hierarchy of similarities between relational structures

q

X = Y (SETTOP 2014) August 21, 2014 6 / 19

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SLIDE 35

Posets of copies of structures

The hierarchy of similarities between relational structures

q

X = Y

q ✟✟✟✟✟✟

X ∼ = Y (SETTOP 2014) August 21, 2014 6 / 19

slide-36
SLIDE 36

Posets of copies of structures

The hierarchy of similarities between relational structures

q

X = Y

q ✟✟✟✟✟✟

X ∼ = Y

q ✟✟✟✟✟✟ X ⇆ Y

(SETTOP 2014) August 21, 2014 6 / 19

slide-37
SLIDE 37

Posets of copies of structures

The hierarchy of similarities between relational structures

q

X = Y

q ✟✟✟✟✟✟

X ∼ = Y

q ✟✟✟✟✟✟ X ⇆ Y q

P(X) = P(Y) (SETTOP 2014) August 21, 2014 6 / 19

slide-38
SLIDE 38

Posets of copies of structures

The hierarchy of similarities between relational structures

q

X = Y

q ✟✟✟✟✟✟

X ∼ = Y

q ✟✟✟✟✟✟ X ⇆ Y q

P(X) = P(Y)

q ✟✟✟✟✟✟

P(X) ∼ = P(Y) (SETTOP 2014) August 21, 2014 6 / 19

slide-39
SLIDE 39

Posets of copies of structures

The hierarchy of similarities between relational structures

q

X = Y

q ✟✟✟✟✟✟

X ∼ = Y

q ✟✟✟✟✟✟ X ⇆ Y q

P(X) = P(Y)

q ✟✟✟✟✟✟

P(X) ∼ = P(Y)

q

sq P(X) ∼ = sq P(Y) (SETTOP 2014) August 21, 2014 6 / 19

slide-40
SLIDE 40

Posets of copies of structures

The hierarchy of similarities between relational structures

q

X = Y

q ✟✟✟✟✟✟

X ∼ = Y

q ✟✟✟✟✟✟ X ⇆ Y q

P(X) = P(Y)

q ✟✟✟✟✟✟

P(X) ∼ = P(Y)

q

sq P(X) ∼ = sq P(Y)

q ✟✟✟✟✟✟

ro sq P(X) ∼ = ro sq P(Y) (SETTOP 2014) August 21, 2014 6 / 19

slide-41
SLIDE 41

Posets of copies of structures

The hierarchy of similarities between relational structures

q

X = Y

q ✟✟✟✟✟✟

X ∼ = Y

q ✟✟✟✟✟✟ X ⇆ Y q

P(X) = P(Y)

q ✟✟✟✟✟✟

P(X) ∼ = P(Y)

q

sq P(X) ∼ = sq P(Y)

q ✟✟✟✟✟✟

ro sq P(X) ∼ = ro sq P(Y)

❍ ❍ ❍ ❍ ❍ ❍

(SETTOP 2014) August 21, 2014 6 / 19

slide-42
SLIDE 42

Posets of copies of structures

The hierarchy of similarities between relational structures

q

X = Y

q ✟✟✟✟✟✟

X ∼ = Y

q ✟✟✟✟✟✟ X ⇆ Y q

P(X) = P(Y)

q ✟✟✟✟✟✟

P(X) ∼ = P(Y)

q

sq P(X) ∼ = sq P(Y)

q ✟✟✟✟✟✟

ro sq P(X) ∼ = ro sq P(Y)

❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍

(SETTOP 2014) August 21, 2014 6 / 19

slide-43
SLIDE 43

Posets of copies of structures

The hierarchy of similarities between relational structures

q

X = Y

q ✟✟✟✟✟✟

X ∼ = Y

q ✟✟✟✟✟✟ X ⇆ Y q

P(X) = P(Y)

q ✟✟✟✟✟✟

P(X) ∼ = P(Y)

q

sq P(X) ∼ = sq P(Y)

q ✟✟✟✟✟✟

ro sq P(X) ∼ = ro sq P(Y)

❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍

(SETTOP 2014) August 21, 2014 6 / 19

slide-44
SLIDE 44

Posets of copies of structures

The hierarchy of similarities between relational structures

q

X = Y

q ✟✟✟✟✟✟

X ∼ = Y

q ✟✟✟✟✟✟ X ⇆ Y q

P(X) = P(Y)

q ✟✟✟✟✟✟

P(X) ∼ = P(Y)

q

sq P(X) ∼ = sq P(Y)

q ✟✟✟✟✟✟

ro sq P(X) ∼ = ro sq P(Y)

❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍

⇔ P(X) ≡ P(Y) (SETTOP 2014) August 21, 2014 6 / 19

slide-45
SLIDE 45

Posets of copies of structures

The hierarchy of similarities between relational structures

q

X = Y

q ✟✟✟✟✟✟

X ∼ = Y

q ✟✟✟✟✟✟ X ⇆ Y q

P(X) = P(Y)

q ✟✟✟✟✟✟

P(X) ∼ = P(Y)

q

sq P(X) ∼ = sq P(Y)

q ✟✟✟✟✟✟

ro sq P(X) ∼ = ro sq P(Y)

❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍

⇔ P(X) ≡ P(Y)

q P(X) = P(Y) ∧ X ∼

= Y (SETTOP 2014) August 21, 2014 6 / 19

slide-46
SLIDE 46

Posets of copies of structures

The hierarchy of similarities between relational structures

q

X = Y

q ✟✟✟✟✟✟

X ∼ = Y

q ✟✟✟✟✟✟ X ⇆ Y q

P(X) = P(Y)

q ✟✟✟✟✟✟

P(X) ∼ = P(Y)

q

sq P(X) ∼ = sq P(Y)

q ✟✟✟✟✟✟

ro sq P(X) ∼ = ro sq P(Y)

❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍

⇔ P(X) ≡ P(Y)

q P(X) = P(Y) ∧ X ∼

= Y

q ✟✟✟✟✟✟

P(X) = P(Y) ∧ X ⇆ Y (SETTOP 2014) August 21, 2014 6 / 19

slide-47
SLIDE 47

Posets of copies of structures

The hierarchy of similarities between relational structures

q

X = Y

q ✟✟✟✟✟✟

X ∼ = Y

q ✟✟✟✟✟✟ X ⇆ Y q

P(X) = P(Y)

q ✟✟✟✟✟✟

P(X) ∼ = P(Y)

q

sq P(X) ∼ = sq P(Y)

q ✟✟✟✟✟✟

ro sq P(X) ∼ = ro sq P(Y)

❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍

⇔ P(X) ≡ P(Y)

q P(X) = P(Y) ∧ X ∼

= Y

q ✟✟✟✟✟✟

P(X) = P(Y) ∧ X ⇆ Y

q P(X) ∼

= P(Y) ∧ X ⇆ Y (SETTOP 2014) August 21, 2014 6 / 19

slide-48
SLIDE 48

Posets of copies of structures

The hierarchy of similarities between relational structures

q

X = Y

q ✟✟✟✟✟✟

X ∼ = Y

q ✟✟✟✟✟✟ X ⇆ Y q

P(X) = P(Y)

q ✟✟✟✟✟✟

P(X) ∼ = P(Y)

q

sq P(X) ∼ = sq P(Y)

q ✟✟✟✟✟✟

ro sq P(X) ∼ = ro sq P(Y)

❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍

⇔ P(X) ≡ P(Y)

q P(X) = P(Y) ∧ X ∼

= Y

q ✟✟✟✟✟✟

P(X) = P(Y) ∧ X ⇆ Y

q P(X) ∼

= P(Y) ∧ X ⇆ Y

q ❍ ❍ ❍ ❍ ❍ ❍

sq P(X) ∼ = sq P(Y) ∧ X ⇆ Y (SETTOP 2014) August 21, 2014 6 / 19

slide-49
SLIDE 49

Posets of copies of structures

The hierarchy of similarities between relational structures

q

X = Y

q ✟✟✟✟✟✟

X ∼ = Y

q ✟✟✟✟✟✟ X ⇆ Y q

P(X) = P(Y)

q ✟✟✟✟✟✟

P(X) ∼ = P(Y)

q

sq P(X) ∼ = sq P(Y)

q ✟✟✟✟✟✟

ro sq P(X) ∼ = ro sq P(Y)

❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍

⇔ P(X) ≡ P(Y)

q P(X) = P(Y) ∧ X ∼

= Y

q ✟✟✟✟✟✟

P(X) = P(Y) ∧ X ⇆ Y

q P(X) ∼

= P(Y) ∧ X ⇆ Y

q ❍ ❍ ❍ ❍ ❍ ❍

sq P(X) ∼ = sq P(Y) ∧ X ⇆ Y a b c d e f g h i j k l m n (SETTOP 2014) August 21, 2014 6 / 19

slide-50
SLIDE 50

Posets of copies of structures

Example: Countable scattered l. o.’ s are in Column D

(SETTOP 2014) August 21, 2014 7 / 19

slide-51
SLIDE 51

Posets of copies of structures

Example: Countable scattered l. o.’ s are in Column D

Theorem ([5]) For each countable scattered linear order X

  • The poset sqP(X), ⊂ is atomless and σ-closed

(SETTOP 2014) August 21, 2014 7 / 19

slide-52
SLIDE 52

Posets of copies of structures

Example: Countable scattered l. o.’ s are in Column D

Theorem ([5]) For each countable scattered linear order X

  • The poset sqP(X), ⊂ is atomless and σ-closed
  • Under CH we have P(X), ⊂ ≡ (P(ω)/ Fin)+.

(SETTOP 2014) August 21, 2014 7 / 19

slide-53
SLIDE 53

Posets of copies of structures

Sub-example: Countable ordinals

(SETTOP 2014) August 21, 2014 8 / 19

slide-54
SLIDE 54

Posets of copies of structures

Sub-example: Countable ordinals

Theorem ([6]) If α = ωγn+rnsn + · · · + ωγ0+r0s0 + k is a countably infinite ordinal presented in the Cantor normal form, then

(SETTOP 2014) August 21, 2014 8 / 19

slide-55
SLIDE 55

Posets of copies of structures

Sub-example: Countable ordinals

Theorem ([6]) If α = ωγn+rnsn + · · · + ωγ0+r0s0 + k is a countably infinite ordinal presented in the Cantor normal form, then sqP(α), ⊂ ∼ =

n

  • i=0
  • rpri(P(ωγi)/Iωγi)

+si

(SETTOP 2014) August 21, 2014 8 / 19

slide-56
SLIDE 56

Posets of copies of structures

Sub-example: Countable ordinals

Theorem ([6]) If α = ωγn+rnsn + · · · + ωγ0+r0s0 + k is a countably infinite ordinal presented in the Cantor normal form, then sqP(α), ⊂ ∼ =

n

  • i=0
  • rpri(P(ωγi)/Iωγi)

+si P(α), ⊂ ≡ (P(ω)/ Fin)+ if α < ω + ω

(SETTOP 2014) August 21, 2014 8 / 19

slide-57
SLIDE 57

Posets of copies of structures

Sub-example: Countable ordinals

Theorem ([6]) If α = ωγn+rnsn + · · · + ωγ0+r0s0 + k is a countably infinite ordinal presented in the Cantor normal form, then sqP(α), ⊂ ∼ =

n

  • i=0
  • rpri(P(ωγi)/Iωγi)

+si P(α), ⊂ ≡ (P(ω)/ Fin)+ if α < ω + ω (P(ω)/ Fin)+ ∗ π if α ≥ ω + ω where [ω] “π is an ω1-closed, separative atomless forcing”.

(SETTOP 2014) August 21, 2014 8 / 19

slide-58
SLIDE 58

Posets of copies of structures

Example: Countable non-scattered l. o.’s are in Column C

(SETTOP 2014) August 21, 2014 9 / 19

slide-59
SLIDE 59

Posets of copies of structures

Example: Countable non-scattered l. o.’s are in Column C

Theorem (with S. Todorˇ cevi´ c, [10]) For each countable non-scattered linear order X we have P(X), ⊂ ≡ S ∗ π where

(SETTOP 2014) August 21, 2014 9 / 19

slide-60
SLIDE 60

Posets of copies of structures

Example: Countable non-scattered l. o.’s are in Column C

Theorem (with S. Todorˇ cevi´ c, [10]) For each countable non-scattered linear order X we have P(X), ⊂ ≡ S ∗ π where

  • S is the Sacks forcing

(SETTOP 2014) August 21, 2014 9 / 19

slide-61
SLIDE 61

Posets of copies of structures

Example: Countable non-scattered l. o.’s are in Column C

Theorem (with S. Todorˇ cevi´ c, [10]) For each countable non-scattered linear order X we have P(X), ⊂ ≡ S ∗ π where

  • S is the Sacks forcing
  • π codes a σ-closed forcing

(SETTOP 2014) August 21, 2014 9 / 19

slide-62
SLIDE 62

Posets of copies of structures

Example: Countable non-scattered l. o.’s are in Column C

Theorem (with S. Todorˇ cevi´ c, [10]) For each countable non-scattered linear order X we have P(X), ⊂ ≡ S ∗ π where

  • S is the Sacks forcing
  • π codes a σ-closed forcing
  • 1S π ≡ (P(ω)/ Fin)+, under CH or PFA.

(SETTOP 2014) August 21, 2014 9 / 19

slide-63
SLIDE 63

Posets of copies of structures

Countable linear orders in the A1 − D5 diagram

scattered

  • l. o.’s

non-scatt.

  • l. o.’s

Q ω ω · ω ω + ω C4 D3 D4 D5 (SETTOP 2014) August 21, 2014 10 / 19

slide-64
SLIDE 64

The monoid of self-embeddings

The poset P(X) and the monoid Emb(X)

(SETTOP 2014) August 21, 2014 11 / 19

slide-65
SLIDE 65

The monoid of self-embeddings

The poset P(X) and the monoid Emb(X)

If M = M, ·, e is a monoid (a semigroup with unity), the right Green’s preorder on M is defined by

(SETTOP 2014) August 21, 2014 11 / 19

slide-66
SLIDE 66

The monoid of self-embeddings

The poset P(X) and the monoid Emb(X)

If M = M, ·, e is a monoid (a semigroup with unity), the right Green’s preorder on M is defined by x R y ⇔ ∃z ∈ M xz = y

(SETTOP 2014) August 21, 2014 11 / 19

slide-67
SLIDE 67

The monoid of self-embeddings

The poset P(X) and the monoid Emb(X)

If M = M, ·, e is a monoid (a semigroup with unity), the right Green’s preorder on M is defined by x R y ⇔ ∃z ∈ M xz = y

Fact If X is a relational structure and Emb(X) = Emb(X), ◦, idX the corresponding monoid of self-embeddings of X, then

(SETTOP 2014) August 21, 2014 11 / 19

slide-68
SLIDE 68

The monoid of self-embeddings

The poset P(X) and the monoid Emb(X)

If M = M, ·, e is a monoid (a semigroup with unity), the right Green’s preorder on M is defined by x R y ⇔ ∃z ∈ M xz = y

Fact If X is a relational structure and Emb(X) = Emb(X), ◦, idX the corresponding monoid of self-embeddings of X, then

  • P(X) = {f[X] : f ∈ Emb(X)}

(SETTOP 2014) August 21, 2014 11 / 19

slide-69
SLIDE 69

The monoid of self-embeddings

The poset P(X) and the monoid Emb(X)

If M = M, ·, e is a monoid (a semigroup with unity), the right Green’s preorder on M is defined by x R y ⇔ ∃z ∈ M xz = y

Fact If X is a relational structure and Emb(X) = Emb(X), ◦, idX the corresponding monoid of self-embeddings of X, then

  • P(X) = {f[X] : f ∈ Emb(X)}
  • P(X), ⊂ ∼

= asqEmb(X), (R)−1

(SETTOP 2014) August 21, 2014 11 / 19

slide-70
SLIDE 70

The monoid of self-embeddings

The poset P(X) and the monoid Emb(X)

If M = M, ·, e is a monoid (a semigroup with unity), the right Green’s preorder on M is defined by x R y ⇔ ∃z ∈ M xz = y

Fact If X is a relational structure and Emb(X) = Emb(X), ◦, idX the corresponding monoid of self-embeddings of X, then

  • P(X) = {f[X] : f ∈ Emb(X)}
  • P(X), ⊂ ∼

= asqEmb(X), (R)−1

  • {f ∈ Emb(X) : f is invertible} = {f ∈ Emb(X) : f is regular} = Aut(X)

(SETTOP 2014) August 21, 2014 11 / 19

slide-71
SLIDE 71

The monoid of self-embeddings

The poset P(X) and the monoid Emb(X)

If M = M, ·, e is a monoid (a semigroup with unity), the right Green’s preorder on M is defined by x R y ⇔ ∃z ∈ M xz = y

Fact If X is a relational structure and Emb(X) = Emb(X), ◦, idX the corresponding monoid of self-embeddings of X, then

  • P(X) = {f[X] : f ∈ Emb(X)}
  • P(X), ⊂ ∼

= asqEmb(X), (R)−1

  • {f ∈ Emb(X) : f is invertible} = {f ∈ Emb(X) : f is regular} = Aut(X)
  • {idX} = {f ∈ Emb(X) : f is idempotent}.

(SETTOP 2014) August 21, 2014 11 / 19

slide-72
SLIDE 72

The monoid of self-embeddings

The poset P(X) and the monoid Emb(X)

If M = M, ·, e is a monoid (a semigroup with unity), the right Green’s preorder on M is defined by x R y ⇔ ∃z ∈ M xz = y

Fact If X is a relational structure and Emb(X) = Emb(X), ◦, idX the corresponding monoid of self-embeddings of X, then

  • P(X) = {f[X] : f ∈ Emb(X)}
  • P(X), ⊂ ∼

= asqEmb(X), (R)−1

  • {f ∈ Emb(X) : f is invertible} = {f ∈ Emb(X) : f is regular} = Aut(X)
  • {idX} = {f ∈ Emb(X) : f is idempotent}.

Theorem If X and Y are relational structures, then X ∼ = Y ⇒ Emb(X) ∼ = Emb(Y) ⇒ P(X), ⊂ ∼ = P(Y), ⊂

(SETTOP 2014) August 21, 2014 11 / 19

slide-73
SLIDE 73

The monoid of self-embeddings

An application of Emb(X) ∼ = Emb(Y) ⇒ P(X), ⊂ ∼ = P(Y), ⊂

(SETTOP 2014) August 21, 2014 12 / 19

slide-74
SLIDE 74

The monoid of self-embeddings

An application of Emb(X) ∼ = Emb(Y) ⇒ P(X), ⊂ ∼ = P(Y), ⊂

Since P((0, 1)Q, <) ∼ = P([0, 1]Q, <) we have Emb((0, 1)Q, <) ∼ = Emb([0, 1]Q, <).

(SETTOP 2014) August 21, 2014 12 / 19

slide-75
SLIDE 75

The monoid of self-embeddings

An application of Emb(X) ∼ = Emb(Y) ⇒ P(X), ⊂ ∼ = P(Y), ⊂

Since P((0, 1)Q, <) ∼ = P([0, 1]Q, <) we have Emb((0, 1)Q, <) ∼ = Emb([0, 1]Q, <). (Comment: but sq P((0, 1)Q, <) ∼ = sq P([0, 1]Q, <))

(SETTOP 2014) August 21, 2014 12 / 19

slide-76
SLIDE 76

The monoid of self-embeddings

Cancellativity commutativity and reversibility of Emb(X)

(SETTOP 2014) August 21, 2014 13 / 19

slide-77
SLIDE 77

The monoid of self-embeddings

Cancellativity commutativity and reversibility of Emb(X)

A monoid M = M, ·, e is (SETTOP 2014) August 21, 2014 13 / 19

slide-78
SLIDE 78

The monoid of self-embeddings

Cancellativity commutativity and reversibility of Emb(X)

A monoid M = M, ·, e is left reversible ⇔ ∀x, y ∃u, v xu = yv (SETTOP 2014) August 21, 2014 13 / 19

slide-79
SLIDE 79

The monoid of self-embeddings

Cancellativity commutativity and reversibility of Emb(X)

A monoid M = M, ·, e is left reversible ⇔ ∀x, y ∃u, v xu = yv right reversible ⇔ ∀x, y ∃u, v ux = vy (SETTOP 2014) August 21, 2014 13 / 19

slide-80
SLIDE 80

The monoid of self-embeddings

Cancellativity commutativity and reversibility of Emb(X)

A monoid M = M, ·, e is left reversible ⇔ ∀x, y ∃u, v xu = yv right reversible ⇔ ∀x, y ∃u, v ux = vy If X is a relational structure, a set A ⊂ X will be called embedding-dense, we will write A ∈ EDense(X) iff (SETTOP 2014) August 21, 2014 13 / 19

slide-81
SLIDE 81

The monoid of self-embeddings

Cancellativity commutativity and reversibility of Emb(X)

A monoid M = M, ·, e is left reversible ⇔ ∀x, y ∃u, v xu = yv right reversible ⇔ ∀x, y ∃u, v ux = vy If X is a relational structure, a set A ⊂ X will be called embedding-dense, we will write A ∈ EDense(X) iff ∀g, h ∈ Emb(X) (g ↾ A = h ↾ A ⇒ g = h). (SETTOP 2014) August 21, 2014 13 / 19

slide-82
SLIDE 82

The monoid of self-embeddings

Cancellativity commutativity and reversibility of Emb(X)

A monoid M = M, ·, e is left reversible ⇔ ∀x, y ∃u, v xu = yv right reversible ⇔ ∀x, y ∃u, v ux = vy If X is a relational structure, a set A ⊂ X will be called embedding-dense, we will write A ∈ EDense(X) iff ∀g, h ∈ Emb(X) (g ↾ A = h ↾ A ⇒ g = h).

Theorem If X is a relational structure, then

(SETTOP 2014) August 21, 2014 13 / 19

slide-83
SLIDE 83

The monoid of self-embeddings

Cancellativity commutativity and reversibility of Emb(X)

A monoid M = M, ·, e is left reversible ⇔ ∀x, y ∃u, v xu = yv right reversible ⇔ ∀x, y ∃u, v ux = vy If X is a relational structure, a set A ⊂ X will be called embedding-dense, we will write A ∈ EDense(X) iff ∀g, h ∈ Emb(X) (g ↾ A = h ↾ A ⇒ g = h).

Theorem If X is a relational structure, then

  • Emb(X) is cancellative ⇔ P(X) ⊂ EDense(X)

(SETTOP 2014) August 21, 2014 13 / 19

slide-84
SLIDE 84

The monoid of self-embeddings

Cancellativity commutativity and reversibility of Emb(X)

A monoid M = M, ·, e is left reversible ⇔ ∀x, y ∃u, v xu = yv right reversible ⇔ ∀x, y ∃u, v ux = vy If X is a relational structure, a set A ⊂ X will be called embedding-dense, we will write A ∈ EDense(X) iff ∀g, h ∈ Emb(X) (g ↾ A = h ↾ A ⇒ g = h).

Theorem If X is a relational structure, then

  • Emb(X) is cancellative ⇔ P(X) ⊂ EDense(X)
  • Emb(X) is left reversible ⇔ the poset P(X), ⊂ is atomic (Column A)

(SETTOP 2014) August 21, 2014 13 / 19

slide-85
SLIDE 85

The monoid of self-embeddings

Cancellativity commutativity and reversibility of Emb(X)

A monoid M = M, ·, e is left reversible ⇔ ∀x, y ∃u, v xu = yv right reversible ⇔ ∀x, y ∃u, v ux = vy If X is a relational structure, a set A ⊂ X will be called embedding-dense, we will write A ∈ EDense(X) iff ∀g, h ∈ Emb(X) (g ↾ A = h ↾ A ⇒ g = h).

Theorem If X is a relational structure, then

  • Emb(X) is cancellative ⇔ P(X) ⊂ EDense(X)
  • Emb(X) is left reversible ⇔ the poset P(X), ⊂ is atomic (Column A)
  • Emb(X) is right reversible ⇔ X has the amalgamation property for

embeddings

(SETTOP 2014) August 21, 2014 13 / 19

slide-86
SLIDE 86

The monoid of self-embeddings

Cancellativity commutativity and reversibility of Emb(X)

A monoid M = M, ·, e is left reversible ⇔ ∀x, y ∃u, v xu = yv right reversible ⇔ ∀x, y ∃u, v ux = vy If X is a relational structure, a set A ⊂ X will be called embedding-dense, we will write A ∈ EDense(X) iff ∀g, h ∈ Emb(X) (g ↾ A = h ↾ A ⇒ g = h).

Theorem If X is a relational structure, then

  • Emb(X) is cancellative ⇔ P(X) ⊂ EDense(X)
  • Emb(X) is left reversible ⇔ the poset P(X), ⊂ is atomic (Column A)
  • Emb(X) is right reversible ⇔ X has the amalgamation property for

embeddings

  • Emb(X) is commutative ⇒ Emb(X) is cancellative, left reversible, and

right reversible.

(SETTOP 2014) August 21, 2014 13 / 19

slide-87
SLIDE 87

The monoid of self-embeddings

Embeddability of Emb(X) into a group

(SETTOP 2014) August 21, 2014 14 / 19

slide-88
SLIDE 88

The monoid of self-embeddings

Embeddability of Emb(X) into a group

Theorem Emb(X) is a (retract of a) group ⇔ P(X) = {X}

(SETTOP 2014) August 21, 2014 14 / 19

slide-89
SLIDE 89

The monoid of self-embeddings

Embeddability of Emb(X) into a group

Theorem Emb(X) is a (retract of a) group ⇔ P(X) = {X} Theorem Each of the following conditions implies that Emb(X) embeds into a group

(SETTOP 2014) August 21, 2014 14 / 19

slide-90
SLIDE 90

The monoid of self-embeddings

Embeddability of Emb(X) into a group

Theorem Emb(X) is a (retract of a) group ⇔ P(X) = {X} Theorem Each of the following conditions implies that Emb(X) embeds into a group

  • Emb(X) is commutative

(SETTOP 2014) August 21, 2014 14 / 19

slide-91
SLIDE 91

The monoid of self-embeddings

Embeddability of Emb(X) into a group

Theorem Emb(X) is a (retract of a) group ⇔ P(X) = {X} Theorem Each of the following conditions implies that Emb(X) embeds into a group

  • Emb(X) is commutative
  • P(X) ⊂ EDense(X) and P(X) is atomic

(SETTOP 2014) August 21, 2014 14 / 19

slide-92
SLIDE 92

The monoid of self-embeddings

Embeddability of Emb(X) into a group

Theorem Emb(X) is a (retract of a) group ⇔ P(X) = {X} Theorem Each of the following conditions implies that Emb(X) embeds into a group

  • Emb(X) is commutative
  • P(X) ⊂ EDense(X) and P(X) is atomic
  • P(X) ⊂ EDense(X) and X has amalgamation for embeddings

(SETTOP 2014) August 21, 2014 14 / 19

slide-93
SLIDE 93

The monoid of self-embeddings

Embeddability of Emb(X) into a group

Theorem Emb(X) is a (retract of a) group ⇔ P(X) = {X} Theorem Each of the following conditions implies that Emb(X) embeds into a group

  • Emb(X) is commutative
  • P(X) ⊂ EDense(X) and P(X) is atomic
  • P(X) ⊂ EDense(X) and X has amalgamation for embeddings
  • Proof. Using theorems of Grothendieck, Ore, and Dubreil.

(SETTOP 2014) August 21, 2014 14 / 19

slide-94
SLIDE 94

The monoid of self-embeddings

Everything is possible

(SETTOP 2014) August 21, 2014 15 / 19

slide-95
SLIDE 95

The monoid of self-embeddings

Everything is possible

X Emb(X) a group Emb(X) commutative Emb(X) embeddable into a group

(SETTOP 2014) August 21, 2014 15 / 19

slide-96
SLIDE 96

The monoid of self-embeddings

Everything is possible

X Emb(X) a group Emb(X) commutative Emb(X) embeddable into a group GZ + +

  • f course

(SETTOP 2014) August 21, 2014 15 / 19

slide-97
SLIDE 97

The monoid of self-embeddings

Everything is possible

X Emb(X) a group Emb(X) commutative Emb(X) embeddable into a group GZ + +

  • f course
  • n≥3 Cn

+

  • f course

(SETTOP 2014) August 21, 2014 15 / 19

slide-98
SLIDE 98

The monoid of self-embeddings

Everything is possible

X Emb(X) a group Emb(X) commutative Emb(X) embeddable into a group GZ + +

  • f course
  • n≥3 Cn

+

  • f course

Gω

  • +

+

(SETTOP 2014) August 21, 2014 15 / 19

slide-99
SLIDE 99

The monoid of self-embeddings

Everything is possible

X Emb(X) a group Emb(X) commutative Emb(X) embeddable into a group GZ + +

  • f course
  • n≥3 Cn

+

  • f course

Gω

  • +

+ Gω ∪

n≥3 Cn

  • +

(SETTOP 2014) August 21, 2014 15 / 19

slide-100
SLIDE 100

The monoid of self-embeddings

Everything is possible

X Emb(X) a group Emb(X) commutative Emb(X) embeddable into a group GZ + +

  • f course
  • n≥3 Cn

+

  • f course

Gω

  • +

+ Gω ∪

n≥3 Cn

  • +

ω, <

  • (SETTOP 2014)

August 21, 2014 15 / 19

slide-101
SLIDE 101

Bi-interpretable structures

Bi-interpretable structures

(SETTOP 2014) August 21, 2014 16 / 19

slide-102
SLIDE 102

Bi-interpretable structures

Bi-interpretable structures

Let X and Y be relational structures of languages LX and LY. (SETTOP 2014) August 21, 2014 16 / 19

slide-103
SLIDE 103

Bi-interpretable structures

Bi-interpretable structures

Let X and Y be relational structures of languages LX and LY.

  • An interpretation of X in Y (without parameters) is a triple n, S, f, where

(SETTOP 2014) August 21, 2014 16 / 19

slide-104
SLIDE 104

Bi-interpretable structures

Bi-interpretable structures

Let X and Y be relational structures of languages LX and LY.

  • An interpretation of X in Y (without parameters) is a triple n, S, f, where
  • n ∈ N,

(SETTOP 2014) August 21, 2014 16 / 19

slide-105
SLIDE 105

Bi-interpretable structures

Bi-interpretable structures

Let X and Y be relational structures of languages LX and LY.

  • An interpretation of X in Y (without parameters) is a triple n, S, f, where
  • n ∈ N,
  • S ⊂ Yn,

(SETTOP 2014) August 21, 2014 16 / 19

slide-106
SLIDE 106

Bi-interpretable structures

Bi-interpretable structures

Let X and Y be relational structures of languages LX and LY.

  • An interpretation of X in Y (without parameters) is a triple n, S, f, where
  • n ∈ N,
  • S ⊂ Yn,
  • f : S → X is a surjection,

(SETTOP 2014) August 21, 2014 16 / 19

slide-107
SLIDE 107

Bi-interpretable structures

Bi-interpretable structures

Let X and Y be relational structures of languages LX and LY.

  • An interpretation of X in Y (without parameters) is a triple n, S, f, where
  • n ∈ N,
  • S ⊂ Yn,
  • f : S → X is a surjection,
  • the pre-image of each set definable in X is definable in Y.

(SETTOP 2014) August 21, 2014 16 / 19

slide-108
SLIDE 108

Bi-interpretable structures

Bi-interpretable structures

Let X and Y be relational structures of languages LX and LY.

  • An interpretation of X in Y (without parameters) is a triple n, S, f, where
  • n ∈ N,
  • S ⊂ Yn,
  • f : S → X is a surjection,
  • the pre-image of each set definable in X is definable in Y.

Then we write f : Y X. (SETTOP 2014) August 21, 2014 16 / 19

slide-109
SLIDE 109

Bi-interpretable structures

Bi-interpretable structures

Let X and Y be relational structures of languages LX and LY.

  • An interpretation of X in Y (without parameters) is a triple n, S, f, where
  • n ∈ N,
  • S ⊂ Yn,
  • f : S → X is a surjection,
  • the pre-image of each set definable in X is definable in Y.

Then we write f : Y X.

  • Structures X and Y are bi-interpretable iff there are interpretations

f : Y X and g : X Y such that the compositions f ∗ g and g ∗ f are definable (in X and Y respectively). (SETTOP 2014) August 21, 2014 16 / 19

slide-110
SLIDE 110

Bi-interpretable structures

Bi-interpretable structures

Let X and Y be relational structures of languages LX and LY.

  • An interpretation of X in Y (without parameters) is a triple n, S, f, where
  • n ∈ N,
  • S ⊂ Yn,
  • f : S → X is a surjection,
  • the pre-image of each set definable in X is definable in Y.

Then we write f : Y X.

  • Structures X and Y are bi-interpretable iff there are interpretations

f : Y X and g : X Y such that the compositions f ∗ g and g ∗ f are definable (in X and Y respectively).

Theorem X and Y are quantifier-free bi-interpretable ⇒ Emb(X) ∼ = Emb(Y).

(SETTOP 2014) August 21, 2014 16 / 19

slide-111
SLIDE 111

Bi-interpretable structures

An application: a dichotomy for ultrahomogeneous str.

(SETTOP 2014) August 21, 2014 17 / 19

slide-112
SLIDE 112

Bi-interpretable structures

An application: a dichotomy for ultrahomogeneous str.

The enlargement of a binary structure X = X, ρ is the structure Xe := X, ρe, where (SETTOP 2014) August 21, 2014 17 / 19

slide-113
SLIDE 113

Bi-interpretable structures

An application: a dichotomy for ultrahomogeneous str.

The enlargement of a binary structure X = X, ρ is the structure Xe := X, ρe, where xρey ⇔ xρy ∨ (x = y ∧ ¬xρy ∧ ¬yρx) (SETTOP 2014) August 21, 2014 17 / 19

slide-114
SLIDE 114

Bi-interpretable structures

An application: a dichotomy for ultrahomogeneous str.

The enlargement of a binary structure X = X, ρ is the structure Xe := X, ρe, where xρey ⇔ xρy ∨ (x = y ∧ ¬xρy ∧ ¬yρx)

Theorem (I) For each countable ultrahomogeneous reflexive or irreflexive binary structure X we have

(SETTOP 2014) August 21, 2014 17 / 19

slide-115
SLIDE 115

Bi-interpretable structures

An application: a dichotomy for ultrahomogeneous str.

The enlargement of a binary structure X = X, ρ is the structure Xe := X, ρe, where xρey ⇔ xρy ∨ (x = y ∧ ¬xρy ∧ ¬yρx)

Theorem (I) For each countable ultrahomogeneous reflexive or irreflexive binary structure X we have

  • Either X is biconnected

(SETTOP 2014) August 21, 2014 17 / 19

slide-116
SLIDE 116

Bi-interpretable structures

An application: a dichotomy for ultrahomogeneous str.

The enlargement of a binary structure X = X, ρ is the structure Xe := X, ρe, where xρey ⇔ xρy ∨ (x = y ∧ ¬xρy ∧ ¬yρx)

Theorem (I) For each countable ultrahomogeneous reflexive or irreflexive binary structure X we have

  • Either X is biconnected
  • Or there is an ultrahomogeneous digraph Y (i.e. Y ∈ Cherlin’s list) and a cardinal

2 ≤ κ ≤ ω such that X is isomorphic to one of the following structures:

(SETTOP 2014) August 21, 2014 17 / 19

slide-117
SLIDE 117

Bi-interpretable structures

An application: a dichotomy for ultrahomogeneous str.

The enlargement of a binary structure X = X, ρ is the structure Xe := X, ρe, where xρey ⇔ xρy ∨ (x = y ∧ ¬xρy ∧ ¬yρx)

Theorem (I) For each countable ultrahomogeneous reflexive or irreflexive binary structure X we have

  • Either X is biconnected
  • Or there is an ultrahomogeneous digraph Y (i.e. Y ∈ Cherlin’s list) and a cardinal

2 ≤ κ ≤ ω such that X is isomorphic to one of the following structures:

κ Ye

(SETTOP 2014) August 21, 2014 17 / 19

slide-118
SLIDE 118

Bi-interpretable structures

An application: a dichotomy for ultrahomogeneous str.

The enlargement of a binary structure X = X, ρ is the structure Xe := X, ρe, where xρey ⇔ xρy ∨ (x = y ∧ ¬xρy ∧ ¬yρx)

Theorem (I) For each countable ultrahomogeneous reflexive or irreflexive binary structure X we have

  • Either X is biconnected
  • Or there is an ultrahomogeneous digraph Y (i.e. Y ∈ Cherlin’s list) and a cardinal

2 ≤ κ ≤ ω such that X is isomorphic to one of the following structures:

κ Ye

  • (

κ Ye)c

(SETTOP 2014) August 21, 2014 17 / 19

slide-119
SLIDE 119

Bi-interpretable structures

An application: a dichotomy for ultrahomogeneous str.

The enlargement of a binary structure X = X, ρ is the structure Xe := X, ρe, where xρey ⇔ xρy ∨ (x = y ∧ ¬xρy ∧ ¬yρx)

Theorem (I) For each countable ultrahomogeneous reflexive or irreflexive binary structure X we have

  • Either X is biconnected
  • Or there is an ultrahomogeneous digraph Y (i.e. Y ∈ Cherlin’s list) and a cardinal

2 ≤ κ ≤ ω such that X is isomorphic to one of the following structures:

κ Ye

  • (

κ Ye)c

  • (

κ Ye)re

(SETTOP 2014) August 21, 2014 17 / 19

slide-120
SLIDE 120

Bi-interpretable structures

An application: a dichotomy for ultrahomogeneous str.

The enlargement of a binary structure X = X, ρ is the structure Xe := X, ρe, where xρey ⇔ xρy ∨ (x = y ∧ ¬xρy ∧ ¬yρx)

Theorem (I) For each countable ultrahomogeneous reflexive or irreflexive binary structure X we have

  • Either X is biconnected
  • Or there is an ultrahomogeneous digraph Y (i.e. Y ∈ Cherlin’s list) and a cardinal

2 ≤ κ ≤ ω such that X is isomorphic to one of the following structures:

κ Ye

  • (

κ Ye)c

  • (

κ Ye)re

  • ((

κ Ye)re)c.

(SETTOP 2014) August 21, 2014 17 / 19

slide-121
SLIDE 121

Bi-interpretable structures

An application: a dichotomy for ultrahomogeneous str.

The enlargement of a binary structure X = X, ρ is the structure Xe := X, ρe, where xρey ⇔ xρy ∨ (x = y ∧ ¬xρy ∧ ¬yρx)

Theorem (I) For each countable ultrahomogeneous reflexive or irreflexive binary structure X we have

  • Either X is biconnected
  • Or there is an ultrahomogeneous digraph Y (i.e. Y ∈ Cherlin’s list) and a cardinal

2 ≤ κ ≤ ω such that X is isomorphic to one of the following structures:

κ Ye

  • (

κ Ye)c

  • (

κ Ye)re

  • ((

κ Ye)re)c.

(II) In the second case we have

(SETTOP 2014) August 21, 2014 17 / 19

slide-122
SLIDE 122

Bi-interpretable structures

An application: a dichotomy for ultrahomogeneous str.

The enlargement of a binary structure X = X, ρ is the structure Xe := X, ρe, where xρey ⇔ xρy ∨ (x = y ∧ ¬xρy ∧ ¬yρx)

Theorem (I) For each countable ultrahomogeneous reflexive or irreflexive binary structure X we have

  • Either X is biconnected
  • Or there is an ultrahomogeneous digraph Y (i.e. Y ∈ Cherlin’s list) and a cardinal

2 ≤ κ ≤ ω such that X is isomorphic to one of the following structures:

κ Ye

  • (

κ Ye)c

  • (

κ Ye)re

  • ((

κ Ye)re)c.

(II) In the second case we have

  • Either P(X) ∼

= P(Z)n, for some biconnected Z ∈ Cherlin’s list and n ≥ 2,

(SETTOP 2014) August 21, 2014 17 / 19

slide-123
SLIDE 123

Bi-interpretable structures

An application: a dichotomy for ultrahomogeneous str.

The enlargement of a binary structure X = X, ρ is the structure Xe := X, ρe, where xρey ⇔ xρy ∨ (x = y ∧ ¬xρy ∧ ¬yρx)

Theorem (I) For each countable ultrahomogeneous reflexive or irreflexive binary structure X we have

  • Either X is biconnected
  • Or there is an ultrahomogeneous digraph Y (i.e. Y ∈ Cherlin’s list) and a cardinal

2 ≤ κ ≤ ω such that X is isomorphic to one of the following structures:

κ Ye

  • (

κ Ye)c

  • (

κ Ye)re

  • ((

κ Ye)re)c.

(II) In the second case we have

  • Either P(X) ∼

= P(Z)n, for some biconnected Z ∈ Cherlin’s list and n ≥ 2,

  • Or sq P(X) is an atomless σ-closed poset (Column D) and, hence,

P(X) ≡ (P(ω)/ Fin)+, under CH.

(SETTOP 2014) August 21, 2014 17 / 19

slide-124
SLIDE 124

Bi-interpretable structures

Proof

(I) Using Ramsey’s theorem.

(SETTOP 2014) August 21, 2014 18 / 19

slide-125
SLIDE 125

Bi-interpretable structures

Proof

(I) Using Ramsey’s theorem. (II) If Y =

k Z then

(SETTOP 2014) August 21, 2014 18 / 19

slide-126
SLIDE 126

Bi-interpretable structures

Proof

(I) Using Ramsey’s theorem. (II) If Y =

k Z then

((

m( k Z)e)re)c and mk Z are quantifier free bi-interpretable and, hence,

(SETTOP 2014) August 21, 2014 18 / 19

slide-127
SLIDE 127

Bi-interpretable structures

Proof

(I) Using Ramsey’s theorem. (II) If Y =

k Z then

((

m( k Z)e)re)c and mk Z are quantifier free bi-interpretable and, hence,

P(X) ∼ = P(

mk Z) ∼

= P(Z)mk.

(SETTOP 2014) August 21, 2014 18 / 19

slide-128
SLIDE 128

Bi-interpretable structures

  • B. Balcar, P. Vopˇ

enka, On systems of almost disjoint sets, Bull. Acad. Polon. Sci. S´

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nski, Zhou Hao Xua, The behaviour of ω2∗ under some consequences of Martin’s axiom, General topology and its relations to modern analysis and algebra, V (Prague, 1981), 577-584, Sigma Ser. Pure Math., 3, Heldermann, Berlin, 1983.

  • P. Vopˇ

enka, A. Pultr, Z. Hedrl´ ın, A rigid relation exists on any set, Comment. Math. Univ. Carolinae 6 (1965) 149–155. (SETTOP 2014) August 21, 2014 19 / 19