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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

  1. Posets of copies of structures Countable binary structures [3] ro sq � P ( X ) , ⊂� i s Borel / M ro P ( ω ) / Fin 2 i s o m o r p h i c t o ♣ ♣ ♣ ♣ ♣ u n d e r CH i ♣ ♣ ♣ ♣ n F d i [ ω ] ω i i n D 5 v d e i ♣ ♣ s a t a n i l o C 4 D 4 b l c l l w e h ♣ ♣ e r e A 3 B 3 C 3 D 3 n d i o d ♣ v e t n i ℵ 0 s i s A 2 B 2 e i d e b ♣ l a i G Z e n 1 l A 1 [ ω ] ω ♣ ♣ ♣ ♣ | P ( X ) | I X P ( X ) σ - c l o s e d X a t o m l e s s sq � P ( X ) , ⊂� ♣ ♣ ♣ ℵ 0 > ℵ 0 1 | sq � P ( X ) , ⊂�| ♣ ♣ ♣ ♣ a t o m i c a t o m l e s s � P ( X ) , ⊂� ♣ ♣ ♣ (SETTOP 2014) August 21, 2014 5 / 19

  2. Posets of copies of structures Countable binary structures [3] ro sq � P ( X ) , ⊂� i s Borel / M ro P ( ω ) / Fin 2 i s o m o r p h i c t o ♣ ♣ ♣ ♣ ♣ u n d e r CH i ♣ ♣ ♣ ♣ n F d i [ ω ] ω i i n D 5 v d e i ♣ ♣ s a t a n i l o C 4 D 4 b l c l l w e h ♣ ♣ e r e A 3 B 3 C 3 D 3 n d i o d ♣ v e t G ω n i ℵ 0 s i s A 2 B 2 e i d e b ♣ l a i G Z e n 1 l A 1 [ ω ] ω ♣ ♣ ♣ ♣ | P ( X ) | I X P ( X ) σ - c l o s e d X a t o m l e s s sq � P ( X ) , ⊂� ♣ ♣ ♣ ℵ 0 > ℵ 0 1 | sq � P ( X ) , ⊂�| ♣ ♣ ♣ ♣ a t o m i c a t o m l e s s � P ( X ) , ⊂� ♣ ♣ ♣ (SETTOP 2014) August 21, 2014 5 / 19

  3. Posets of copies of structures Countable binary structures [3] ro sq � P ( X ) , ⊂� i s Borel / M ro P ( ω ) / Fin 2 i s o m o r p h i c t o ♣ ♣ ♣ ♣ ♣ u n d e r CH i ♣ ♣ ♣ ♣ n F d i [ ω ] ω i i n D 5 v d e i ♣ ♣ s a t a n i l o C 4 D 4 b l c l l w e h ♣ ♣ e r e A 3 B 3 C 3 D 3 n d i o d ♣ v e D <ω 2 t G ω n i ℵ 0 s i s A 2 B 2 e i d e b ♣ l a i G Z e n 1 l A 1 [ ω ] ω ♣ ♣ ♣ ♣ | P ( X ) | I X P ( X ) σ - c l o s e d X a t o m l e s s sq � P ( X ) , ⊂� ♣ ♣ ♣ ℵ 0 > ℵ 0 1 | sq � P ( X ) , ⊂�| ♣ ♣ ♣ ♣ a t o m i c a t o m l e s s � P ( X ) , ⊂� ♣ ♣ ♣ (SETTOP 2014) August 21, 2014 5 / 19

  4. Posets of copies of structures Countable binary structures [3] ro sq � P ( X ) , ⊂� i s Borel / M ro P ( ω ) / Fin 2 i s o m o r p h i c t o ♣ ♣ ♣ ♣ ♣ u n d e r CH i ♣ ♣ ♣ ♣ n � ω, < � F d i [ ω ] ω i i n D 5 v d e i ♣ ♣ s a t a n i l o C 4 D 4 b l c l l w e h ♣ ♣ e r e A 3 B 3 C 3 D 3 n d i o d ♣ v e D <ω 2 t G ω n i ℵ 0 s i s A 2 B 2 e i d e b ♣ l a i G Z e n 1 l A 1 [ ω ] ω ♣ ♣ ♣ ♣ | P ( X ) | I X P ( X ) σ - c l o s e d X a t o m l e s s sq � P ( X ) , ⊂� ♣ ♣ ♣ ℵ 0 > ℵ 0 1 | sq � P ( X ) , ⊂�| ♣ ♣ ♣ ♣ a t o m i c a t o m l e s s � P ( X ) , ⊂� ♣ ♣ ♣ (SETTOP 2014) August 21, 2014 5 / 19

  5. Posets of copies of structures Countable binary structures [3] ro sq � P ( X ) , ⊂� i s Borel / M ro P ( ω ) / Fin 2 i s o m o r p h i c t o ♣ ♣ ♣ ♣ ♣ u n d e r CH i ♣ ♣ ♣ ♣ n � ω, < � F d i [ ω ] ω i i n D 5 v d e i ♣ ♣ s a t Q a n i l o C 4 D 4 b l c l l w e h ♣ ♣ e r e A 3 B 3 C 3 D 3 n d i o d ♣ v e D <ω 2 t G ω n i ℵ 0 s i s A 2 B 2 e i d e b ♣ l a i G Z e n 1 l A 1 [ ω ] ω ♣ ♣ ♣ ♣ | P ( X ) | I X P ( X ) σ - c l o s e d X a t o m l e s s sq � P ( X ) , ⊂� ♣ ♣ ♣ ℵ 0 > ℵ 0 1 | sq � P ( X ) , ⊂�| ♣ ♣ ♣ ♣ a t o m i c a t o m l e s s � P ( X ) , ⊂� ♣ ♣ ♣ (SETTOP 2014) August 21, 2014 5 / 19

  6. Posets of copies of structures Countable binary structures [3] ro sq � P ( X ) , ⊂� i s Borel / M ro P ( ω ) / Fin 2 i s o m o r p h i c t o ♣ ♣ ♣ ♣ ♣ u n d e r CH i ♣ ♣ ♣ ♣ n � ω, < � F d i [ ω ] ω i i n D 5 v d e i ♣ ♣ s a t Q a n i l o C 4 D 4 b l c l l w e h ♣ ♣ e X Col ( ω, c ) r e A 3 B 3 C 3 D 3 n d i o d ♣ v e D <ω 2 t G ω n i ℵ 0 s i s A 2 B 2 e i d e b ♣ l a i G Z e n 1 l A 1 [ ω ] ω ♣ ♣ ♣ ♣ | P ( X ) | I X P ( X ) σ - c l o s e d X a t o m l e s s sq � P ( X ) , ⊂� ♣ ♣ ♣ ℵ 0 > ℵ 0 1 | sq � P ( X ) , ⊂�| ♣ ♣ ♣ ♣ a t o m i c a t o m l e s s � P ( X ) , ⊂� ♣ ♣ ♣ (SETTOP 2014) August 21, 2014 5 / 19

  7. Posets of copies of structures The hierarchy of similarities between relational structures (SETTOP 2014) August 21, 2014 6 / 19

  8. Posets of copies of structures The hierarchy of similarities between relational structures q X = Y (SETTOP 2014) August 21, 2014 6 / 19

  9. Posets of copies of structures The hierarchy of similarities between relational structures ✟✟✟✟✟✟ X ∼ = Y q q X = Y (SETTOP 2014) August 21, 2014 6 / 19

  10. Posets of copies of structures The hierarchy of similarities between relational structures ✟✟✟✟✟✟ X ⇆ Y q ✟✟✟✟✟✟ X ∼ = Y q q X = Y (SETTOP 2014) August 21, 2014 6 / 19

  11. Posets of copies of structures The hierarchy of similarities between relational structures ✟✟✟✟✟✟ X ⇆ Y q P ( X ) = P ( Y ) q ✟✟✟✟✟✟ X ∼ = Y q q X = Y (SETTOP 2014) August 21, 2014 6 / 19

  12. Posets of copies of structures The hierarchy of similarities between relational structures ✟✟✟✟✟✟ X ⇆ Y q ✟✟✟✟✟✟ P ( X ) ∼ = P ( Y ) q P ( X ) = P ( Y ) q ✟✟✟✟✟✟ X ∼ = Y q q X = Y (SETTOP 2014) August 21, 2014 6 / 19

  13. Posets of copies of structures The hierarchy of similarities between relational structures ✟✟✟✟✟✟ X ⇆ Y sq P ( X ) ∼ = sq P ( Y ) q q ✟✟✟✟✟✟ P ( X ) ∼ = P ( Y ) q P ( X ) = P ( Y ) q ✟✟✟✟✟✟ X ∼ = Y q q X = Y (SETTOP 2014) August 21, 2014 6 / 19

  14. Posets of copies of structures The hierarchy of similarities between relational structures ro sq P ( X ) ∼ = ro sq P ( Y ) ✟✟✟✟✟✟ q ✟✟✟✟✟✟ X ⇆ Y sq P ( X ) ∼ = sq P ( Y ) q q ✟✟✟✟✟✟ P ( X ) ∼ = P ( Y ) q P ( X ) = P ( Y ) q ✟✟✟✟✟✟ X ∼ = Y q q X = Y (SETTOP 2014) August 21, 2014 6 / 19

  15. Posets of copies of structures The hierarchy of similarities between relational structures ro sq P ( X ) ∼ = ro sq P ( Y ) ✟✟✟✟✟✟ q ✟✟✟✟✟✟ X ⇆ Y sq P ( X ) ∼ = sq P ( Y ) q q ✟✟✟✟✟✟ P ( X ) ∼ = P ( Y ) q ❍ P ( X ) = P ( Y ) q ❍ ❍ ❍ ❍ ❍ ✟✟✟✟✟✟ X ∼ = Y q q X = Y (SETTOP 2014) August 21, 2014 6 / 19

  16. Posets of copies of structures The hierarchy of similarities between relational structures ro sq P ( X ) ∼ = ro sq P ( Y ) ✟✟✟✟✟✟ q ✟✟✟✟✟✟ X ⇆ Y sq P ( X ) ∼ = sq P ( Y ) q q ✟✟✟✟✟✟ P ( X ) ∼ ❍ = P ( Y ) q ❍ ❍ ❍ ❍ ❍ ❍ P ( X ) = P ( Y ) q ❍ ❍ ❍ ❍ ❍ ✟✟✟✟✟✟ X ∼ = Y q q X = Y (SETTOP 2014) August 21, 2014 6 / 19

  17. Posets of copies of structures The hierarchy of similarities between relational structures ro sq P ( X ) ∼ = ro sq P ( Y ) ✟✟✟✟✟✟ ❍ q ❍ ❍ ❍ ❍ ❍ ✟✟✟✟✟✟ X ⇆ Y sq P ( X ) ∼ = sq P ( Y ) q q ✟✟✟✟✟✟ P ( X ) ∼ ❍ = P ( Y ) q ❍ ❍ ❍ ❍ ❍ ❍ P ( X ) = P ( Y ) q ❍ ❍ ❍ ❍ ❍ ✟✟✟✟✟✟ X ∼ = Y q q X = Y (SETTOP 2014) August 21, 2014 6 / 19

  18. Posets of copies of structures The hierarchy of similarities between relational structures ro sq P ( X ) ∼ = ro sq P ( Y ) ⇔ P ( X ) ≡ P ( Y ) ✟✟✟✟✟✟ ❍ q ❍ ❍ ❍ ❍ ❍ ✟✟✟✟✟✟ X ⇆ Y sq P ( X ) ∼ = sq P ( Y ) q q ✟✟✟✟✟✟ P ( X ) ∼ ❍ = P ( Y ) q ❍ ❍ ❍ ❍ ❍ ❍ P ( X ) = P ( Y ) q ❍ ❍ ❍ ❍ ❍ ✟✟✟✟✟✟ X ∼ = Y q q X = Y (SETTOP 2014) August 21, 2014 6 / 19

  19. Posets of copies of structures The hierarchy of similarities between relational structures ro sq P ( X ) ∼ = ro sq P ( Y ) ⇔ P ( X ) ≡ P ( Y ) ✟✟✟✟✟✟ ❍ q ❍ ❍ ❍ ❍ ❍ ✟✟✟✟✟✟ X ⇆ Y sq P ( X ) ∼ = sq P ( Y ) q q ✟✟✟✟✟✟ P ( X ) ∼ ❍ = P ( Y ) q ❍ ❍ ❍ ❍ ❍ ❍ P ( X ) = P ( Y ) q ❍ ❍ ❍ ❍ ❍ ✟✟✟✟✟✟ X ∼ = Y q q P ( X ) = P ( Y ) ∧ X ∼ = Y q X = Y (SETTOP 2014) August 21, 2014 6 / 19

  20. Posets of copies of structures The hierarchy of similarities between relational structures ro sq P ( X ) ∼ = ro sq P ( Y ) ⇔ P ( X ) ≡ P ( Y ) ✟✟✟✟✟✟ ❍ q ❍ ❍ ❍ ❍ ❍ ✟✟✟✟✟✟ X ⇆ Y sq P ( X ) ∼ = sq P ( Y ) q q ✟✟✟✟✟✟ P ( X ) ∼ ❍ = P ( Y ) q ❍ ❍ ❍ ❍ ❍ ❍ ✟✟✟✟✟✟ P ( X ) = P ( Y ) q ❍ ❍ ❍ ❍ ❍ ✟✟✟✟✟✟ X ∼ P ( X ) = P ( Y ) ∧ X ⇆ Y = Y q q q P ( X ) = P ( Y ) ∧ X ∼ = Y q X = Y (SETTOP 2014) August 21, 2014 6 / 19

  21. Posets of copies of structures The hierarchy of similarities between relational structures ro sq P ( X ) ∼ = ro sq P ( Y ) ⇔ P ( X ) ≡ P ( Y ) ✟✟✟✟✟✟ ❍ q ❍ ❍ ❍ ❍ ❍ ✟✟✟✟✟✟ X ⇆ Y sq P ( X ) ∼ = sq P ( Y ) q q ✟✟✟✟✟✟ P ( X ) ∼ ❍ = P ( Y ) q ❍ ❍ ❍ ❍ ❍ ❍ ✟✟✟✟✟✟ q P ( X ) ∼ P ( X ) = P ( Y ) = P ( Y ) ∧ X ⇆ Y q ❍ ❍ ❍ ❍ ❍ ✟✟✟✟✟✟ X ∼ P ( X ) = P ( Y ) ∧ X ⇆ Y = Y q q q P ( X ) = P ( Y ) ∧ X ∼ = Y q X = Y (SETTOP 2014) August 21, 2014 6 / 19

  22. Posets of copies of structures The hierarchy of similarities between relational structures ro sq P ( X ) ∼ = ro sq P ( Y ) ⇔ P ( X ) ≡ P ( Y ) ✟✟✟✟✟✟ ❍ q ❍ ❍ ❍ ❍ ❍ ❍ ✟✟✟✟✟✟ X ⇆ Y sq P ( X ) ∼ = sq P ( Y ) q q ❍ ❍ ❍ ❍ ❍ ✟✟✟✟✟✟ P ( X ) ∼ ❍ sq P ( X ) ∼ = P ( Y ) = sq P ( Y ) ∧ X ⇆ Y q ❍ q ❍ ❍ ❍ ❍ ❍ ✟✟✟✟✟✟ q P ( X ) ∼ P ( X ) = P ( Y ) = P ( Y ) ∧ X ⇆ Y q ❍ ❍ ❍ ❍ ❍ ✟✟✟✟✟✟ X ∼ P ( X ) = P ( Y ) ∧ X ⇆ Y = Y q q q P ( X ) = P ( Y ) ∧ X ∼ = Y q X = Y (SETTOP 2014) August 21, 2014 6 / 19

  23. Posets of copies of structures The hierarchy of similarities between relational structures ro sq P ( X ) ∼ = ro sq P ( Y ) ⇔ P ( X ) ≡ P ( Y ) ✟✟✟✟✟✟ ❍ q ❍ ❍ ❍ m n ❍ ❍ ❍ ✟✟✟✟✟✟ X ⇆ Y sq P ( X ) ∼ = sq P ( Y ) q q ❍ ❍ ❍ j k l ❍ ❍ ✟✟✟✟✟✟ P ( X ) ∼ ❍ sq P ( X ) ∼ = P ( Y ) = sq P ( Y ) ∧ X ⇆ Y q ❍ q ❍ ❍ g h ❍ i ❍ ❍ ✟✟✟✟✟✟ q P ( X ) ∼ P ( X ) = P ( Y ) = P ( Y ) ∧ X ⇆ Y q ❍ ❍ ❍ d e f ❍ ❍ ✟✟✟✟✟✟ X ∼ P ( X ) = P ( Y ) ∧ X ⇆ Y = Y q q c b q P ( X ) = P ( Y ) ∧ X ∼ = Y a q X = Y (SETTOP 2014) August 21, 2014 6 / 19

  24. Posets of copies of structures Example: Countable scattered l. o.’ s are in Column D (SETTOP 2014) August 21, 2014 7 / 19

  25. 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 sq � P ( X ) , ⊂� is atomless and σ -closed (SETTOP 2014) August 21, 2014 7 / 19

  26. 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 sq � P ( X ) , ⊂� is atomless and σ -closed • Under CH we have � P ( X ) , ⊂� ≡ ( P ( ω ) / Fin ) + . (SETTOP 2014) August 21, 2014 7 / 19

  27. Posets of copies of structures Sub-example: Countable ordinals (SETTOP 2014) August 21, 2014 8 / 19

  28. Posets of copies of structures Sub-example: Countable ordinals Theorem ([6]) If α = ω γ n + r n s n + · · · + ω γ 0 + r 0 s 0 + k is a countably infinite ordinal presented in the Cantor normal form, then (SETTOP 2014) August 21, 2014 8 / 19

  29. Posets of copies of structures Sub-example: Countable ordinals Theorem ([6]) If α = ω γ n + r n s n + · · · + ω γ 0 + r 0 s 0 + k is a countably infinite ordinal presented in the Cantor normal form, then n � + � s i �� sq � P ( α ) , ⊂� ∼ � rp r i ( P ( ω γ i ) / I ω γ i ) = i = 0 (SETTOP 2014) August 21, 2014 8 / 19

  30. Posets of copies of structures Sub-example: Countable ordinals Theorem ([6]) If α = ω γ n + r n s n + · · · + ω γ 0 + r 0 s 0 + k is a countably infinite ordinal presented in the Cantor normal form, then n � + � s i �� sq � P ( α ) , ⊂� ∼ � rp r i ( P ( ω γ i ) / I ω γ i ) = i = 0 � ( P ( ω ) / Fin ) + if α < ω + ω � P ( α ) , ⊂� ≡ (SETTOP 2014) August 21, 2014 8 / 19

  31. Posets of copies of structures Sub-example: Countable ordinals Theorem ([6]) If α = ω γ n + r n s n + · · · + ω γ 0 + r 0 s 0 + k is a countably infinite ordinal presented in the Cantor normal form, then n � + � s i �� sq � P ( α ) , ⊂� ∼ � rp r i ( P ( ω γ i ) / I ω γ i ) = i = 0 � ( P ( ω ) / Fin ) + if α < ω + ω � P ( α ) , ⊂� ≡ ( P ( ω ) / Fin ) + ∗ π if α ≥ ω + ω where [ ω ] � “ π is an ω 1 -closed, separative atomless forcing”. (SETTOP 2014) August 21, 2014 8 / 19

  32. Posets of copies of structures Example: Countable non-scattered l. o.’s are in Column C (SETTOP 2014) August 21, 2014 9 / 19

  33. 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

  34. 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

  35. 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

  36. 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 • 1 S � π ≡ ( P ( ω ) / Fin ) + , under CH or PFA. (SETTOP 2014) August 21, 2014 9 / 19

  37. Posets of copies of structures Countable linear orders in the A 1 − D 5 diagram scattered l. o.’s ω D 5 non-scatt. l. o.’s Q ω · ω C 4 D 4 ω + ω D 3 (SETTOP 2014) August 21, 2014 10 / 19

  38. The monoid of self-embeddings The poset P ( X ) and the monoid E mb ( X ) (SETTOP 2014) August 21, 2014 11 / 19

  39. The monoid of self-embeddings The poset P ( X ) and the monoid E mb ( 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

  40. The monoid of self-embeddings The poset P ( X ) and the monoid E mb ( 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

  41. The monoid of self-embeddings The poset P ( X ) and the monoid E mb ( 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 E mb ( X ) = � Emb ( X ) , ◦ , id X � the corresponding monoid of self-embeddings of X , then (SETTOP 2014) August 21, 2014 11 / 19

  42. The monoid of self-embeddings The poset P ( X ) and the monoid E mb ( 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 E mb ( X ) = � Emb ( X ) , ◦ , id X � the corresponding monoid of self-embeddings of X , then • P ( X ) = { f [ X ] : f ∈ Emb ( X ) } (SETTOP 2014) August 21, 2014 11 / 19

  43. The monoid of self-embeddings The poset P ( X ) and the monoid E mb ( 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 E mb ( X ) = � Emb ( X ) , ◦ , id X � the corresponding monoid of self-embeddings of X , then • P ( X ) = { f [ X ] : f ∈ Emb ( X ) } • � P ( X ) , ⊂� ∼ = asq � Emb ( X ) , ( � R ) − 1 � (SETTOP 2014) August 21, 2014 11 / 19

  44. The monoid of self-embeddings The poset P ( X ) and the monoid E mb ( 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 E mb ( X ) = � Emb ( X ) , ◦ , id X � the corresponding monoid of self-embeddings of X , then • P ( X ) = { f [ X ] : f ∈ Emb ( X ) } • � P ( X ) , ⊂� ∼ = asq � Emb ( X ) , ( � R ) − 1 � • { f ∈ Emb ( X ) : f is invertible } = { f ∈ Emb ( X ) : f is regular } = Aut ( X ) (SETTOP 2014) August 21, 2014 11 / 19

  45. The monoid of self-embeddings The poset P ( X ) and the monoid E mb ( 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 E mb ( X ) = � Emb ( X ) , ◦ , id X � the corresponding monoid of self-embeddings of X , then • P ( X ) = { f [ X ] : f ∈ Emb ( X ) } • � P ( X ) , ⊂� ∼ = asq � Emb ( X ) , ( � R ) − 1 � • { f ∈ Emb ( X ) : f is invertible } = { f ∈ Emb ( X ) : f is regular } = Aut ( X ) • { id X } = { f ∈ Emb ( X ) : f is idempotent } . (SETTOP 2014) August 21, 2014 11 / 19

  46. The monoid of self-embeddings The poset P ( X ) and the monoid E mb ( 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 E mb ( X ) = � Emb ( X ) , ◦ , id X � the corresponding monoid of self-embeddings of X , then • P ( X ) = { f [ X ] : f ∈ Emb ( X ) } • � P ( X ) , ⊂� ∼ = asq � Emb ( X ) , ( � R ) − 1 � • { f ∈ Emb ( X ) : f is invertible } = { f ∈ Emb ( X ) : f is regular } = Aut ( X ) • { id X } = { f ∈ Emb ( X ) : f is idempotent } . Theorem If X and Y are relational structures, then X ∼ = Y ⇒ E mb ( X ) ∼ = E mb ( Y ) ⇒ � P ( X ) , ⊂� ∼ = � P ( Y ) , ⊂� (SETTOP 2014) August 21, 2014 11 / 19

  47. The monoid of self-embeddings An application of E mb ( X ) ∼ = E mb ( Y ) ⇒ � P ( X ) , ⊂� ∼ = � P ( Y ) , ⊂� (SETTOP 2014) August 21, 2014 12 / 19

  48. The monoid of self-embeddings An application of E mb ( X ) ∼ = E mb ( Y ) ⇒ � P ( X ) , ⊂� ∼ = � P ( Y ) , ⊂� Since P (( 0 , 1 ) Q , < ) �∼ = P ([ 0 , 1 ] Q , < ) we have E mb (( 0 , 1 ) Q , < ) �∼ = E mb ([ 0 , 1 ] Q , < ) . (SETTOP 2014) August 21, 2014 12 / 19

  49. The monoid of self-embeddings An application of E mb ( X ) ∼ = E mb ( Y ) ⇒ � P ( X ) , ⊂� ∼ = � P ( Y ) , ⊂� Since P (( 0 , 1 ) Q , < ) �∼ = P ([ 0 , 1 ] Q , < ) we have E mb (( 0 , 1 ) Q , < ) �∼ = E mb ([ 0 , 1 ] Q , < ) . (Comment: but sq P (( 0 , 1 ) Q , < ) ∼ = sq P ([ 0 , 1 ] Q , < ) ) (SETTOP 2014) August 21, 2014 12 / 19

  50. The monoid of self-embeddings Cancellativity commutativity and reversibility of Emb ( X ) (SETTOP 2014) August 21, 2014 13 / 19

  51. 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

  52. 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

  53. 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

  54. 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

  55. 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

  56. 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

  57. 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 • E mb ( X ) is cancellative ⇔ P ( X ) ⊂ EDense ( X ) (SETTOP 2014) August 21, 2014 13 / 19

  58. 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 • E mb ( X ) is cancellative ⇔ P ( X ) ⊂ EDense ( X ) • E mb ( X ) is left reversible ⇔ the poset � P ( X ) , ⊂� is atomic (Column A) (SETTOP 2014) August 21, 2014 13 / 19

  59. 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 • E mb ( X ) is cancellative ⇔ P ( X ) ⊂ EDense ( X ) • E mb ( X ) is left reversible ⇔ the poset � P ( X ) , ⊂� is atomic (Column A) • E mb ( X ) is right reversible ⇔ X has the amalgamation property for embeddings (SETTOP 2014) August 21, 2014 13 / 19

  60. 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 • E mb ( X ) is cancellative ⇔ P ( X ) ⊂ EDense ( X ) • E mb ( X ) is left reversible ⇔ the poset � P ( X ) , ⊂� is atomic (Column A) • E mb ( X ) is right reversible ⇔ X has the amalgamation property for embeddings • E mb ( X ) is commutative ⇒ E mb ( X ) is cancellative, left reversible, and right reversible. (SETTOP 2014) August 21, 2014 13 / 19

  61. The monoid of self-embeddings Embeddability of Emb ( X ) into a group (SETTOP 2014) August 21, 2014 14 / 19

  62. The monoid of self-embeddings Embeddability of Emb ( X ) into a group Theorem E mb ( X ) is a (retract of a) group ⇔ P ( X ) = { X } (SETTOP 2014) August 21, 2014 14 / 19

  63. The monoid of self-embeddings Embeddability of Emb ( X ) into a group Theorem E mb ( X ) is a (retract of a) group ⇔ P ( X ) = { X } Theorem Each of the following conditions implies that E mb ( X ) embeds into a group (SETTOP 2014) August 21, 2014 14 / 19

  64. The monoid of self-embeddings Embeddability of Emb ( X ) into a group Theorem E mb ( X ) is a (retract of a) group ⇔ P ( X ) = { X } Theorem Each of the following conditions implies that E mb ( X ) embeds into a group • E mb ( X ) is commutative (SETTOP 2014) August 21, 2014 14 / 19

  65. The monoid of self-embeddings Embeddability of Emb ( X ) into a group Theorem E mb ( X ) is a (retract of a) group ⇔ P ( X ) = { X } Theorem Each of the following conditions implies that E mb ( X ) embeds into a group • E mb ( X ) is commutative • P ( X ) ⊂ EDense ( X ) and P ( X ) is atomic (SETTOP 2014) August 21, 2014 14 / 19

  66. The monoid of self-embeddings Embeddability of Emb ( X ) into a group Theorem E mb ( X ) is a (retract of a) group ⇔ P ( X ) = { X } Theorem Each of the following conditions implies that E mb ( X ) embeds into a group • E mb ( 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

  67. The monoid of self-embeddings Embeddability of Emb ( X ) into a group Theorem E mb ( X ) is a (retract of a) group ⇔ P ( X ) = { X } Theorem Each of the following conditions implies that E mb ( X ) embeds into a group • E mb ( 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

  68. The monoid of self-embeddings Everything is possible (SETTOP 2014) August 21, 2014 15 / 19

  69. The monoid of self-embeddings Everything is possible X E mb ( X ) a group E mb ( X ) commutative E mb ( X ) embeddable into a group (SETTOP 2014) August 21, 2014 15 / 19

  70. The monoid of self-embeddings Everything is possible X E mb ( X ) a group E mb ( X ) commutative E mb ( X ) embeddable into a group G Z + + of course (SETTOP 2014) August 21, 2014 15 / 19

  71. The monoid of self-embeddings Everything is possible X E mb ( X ) a group E mb ( X ) commutative E mb ( X ) embeddable into a group G Z + + of course � n ≥ 3 C n + - of course (SETTOP 2014) August 21, 2014 15 / 19

  72. The monoid of self-embeddings Everything is possible X E mb ( X ) a group E mb ( X ) commutative E mb ( X ) embeddable into a group G Z + + of course � n ≥ 3 C n + - of course G ω - + + (SETTOP 2014) August 21, 2014 15 / 19

  73. The monoid of self-embeddings Everything is possible X E mb ( X ) a group E mb ( X ) commutative E mb ( X ) embeddable into a group G Z + + of course � n ≥ 3 C n + - of course G ω - + + G ω ∪ � n ≥ 3 C n - - + (SETTOP 2014) August 21, 2014 15 / 19

  74. The monoid of self-embeddings Everything is possible X E mb ( X ) a group E mb ( X ) commutative E mb ( X ) embeddable into a group G Z + + of course � n ≥ 3 C n + - of course G ω - + + G ω ∪ � n ≥ 3 C n - - + � ω, < � - - - (SETTOP 2014) August 21, 2014 15 / 19

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