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H OW TO FIND A FINITE ALGEBRA WITH A GIVEN CONGRUENCE LATTICE ? H OW - PowerPoint PPT Presentation

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S MALL C ONGRUENCE L ATTICES William DeMeo williamdemeo@gmail.com University of South Carolina joint work with Ralph Freese, Peter Jipsen, Bill Lampe, J.B. Nation B LA ST Conference Chapman University August 59, 2013 T HE P ROBLEM C

  1. F INDING REPRESENTATIONS ... ... AS INTERVALS IN SUBGROUP LATTICES G L 17 The group G = ( A 4 × A 4 ) ⋊ C 2 has a subgroup H ∼ = S 3 such that � H , G � ∼ = L 17 . ...so the dual L 16 is also representable. H SmallGroup(288,1025) | G : H | = 48 L 13

  2. F INDING REPRESENTATIONS ... ... AS INTERVALS IN SUBGROUP LATTICES G L 17 The group G = ( A 4 × A 4 ) ⋊ C 2 has a subgroup H ∼ = S 3 such that � H , G � ∼ = L 17 . ...so the dual L 16 is also representable. H SmallGroup(288,1025) | G : H | = 48 G L 13 The group G = ( C 2 × C 2 × C 2 × C 2 ) ⋊ A 5 has a subgroup H ∼ = A 4 such that � H , G � ∼ = L 13 . H SmallGroup(960,11358) | G : H | = 80

  3. A RE ALL LATTICES WITH AT MOST 7 ELEMENTS REPRESENTABLE ? � L 20 � L 19 L 11 L 13 � L 17 � L 10 L 9

  4. F INDING REPRESENTATIONS ... ... USING SUBGROUP LATTICE INTERVALS AND THE FILTER + IDEAL LEMMA . L 11

  5. F INDING REPRESENTATIONS ... ... USING SUBGROUP LATTICE INTERVALS AND THE FILTER + IDEAL LEMMA . SmallGroup(288,1025) Let G = ( A 4 × A 4 ) ⋊ C 2 . G G has a subgroup H ∼ = C 6 with � H , G � ∼ = N 5 . Let � H , G � = { H , α, β, γ, G } ∼ L 11 = N 5 . β α γ H | G : H | = 48 1

  6. F INDING REPRESENTATIONS ... ... USING SUBGROUP LATTICE INTERVALS AND THE FILTER + IDEAL LEMMA . SmallGroup(288,1025) Let G = ( A 4 × A 4 ) ⋊ C 2 . G G has a subgroup H ∼ = C 6 with � H , G � ∼ = N 5 . Let � H , G � = { H , α, β, γ, G } ∼ L 11 = N 5 . β α Sub ( G ) is a congruence lattice, so if there exists a subgroup K ≻ 1 , below β and not γ K below γ , then H = K ↓ ∪ H ↑ . | G : H | = 48 L 11 ∼ 1

  7. F INDING REPRESENTATIONS ... ... USING SUBGROUP LATTICE INTERVALS AND THE FILTER + IDEAL LEMMA . SmallGroup(288,1025) Let G = ( A 4 × A 4 ) ⋊ C 2 . G G has a subgroup H ∼ = C 6 with � H , G � ∼ = N 5 . Let � H , G � = { H , α, β, γ, G } ∼ L 11 = N 5 . β α Sub ( G ) is a congruence lattice, so if there exists a subgroup K ≻ 1 , below β and not γ K below γ , then H = K ↓ ∪ H ↑ . | G : H | = 48 L 11 ∼ 1 Sub ( A 4 ) is a congruence lattice A 4 (of A 4 acting regularly on itself). L 17 Therefore, V 4 L 17 ∼ = V ↓ 4 ∪ P ↑ P is a congruence lattice.

  8. A RE ALL LATTICES WITH AT MOST 7 ELEMENTS REPRESENTABLE ? � L 20 � L 19 L 11 � L 13 � L 17 � L 10 L 9

  9. A RE ALL LATTICES WITH AT MOST 7 ELEMENTS REPRESENTABLE ? � L 20 � L 19 L 11 � L 13 � L 17 � � L 10 L 9

  10. C ONSTRUCTION OF AN ALGEBRA A WITH Con A ∼ = L 9 . M 4 L 9

  11. C ONSTRUCTION OF AN ALGEBRA A WITH Con A ∼ = L 9 . S TEP 1 Take a permutational algebra B = � B , F � with congruence lattice Con B ∼ = M 4 . 1 B γ α β δ Con B 0 B L 9

  12. C ONSTRUCTION OF AN ALGEBRA A WITH Con A ∼ = L 9 . S TEP 1 Take a permutational algebra B = � B , F � with congruence lattice Con B ∼ = M 4 . Example: 1 B Let B = { 0 , 1 , . . . , 5 } index the elements of S 3 and consider the right regular action of S 3 on itself. γ α β g 0 = ( 0 , 4 )( 1 , 3 )( 2 , 5 ) and g 1 = ( 0 , 1 , 2 )( 3 , 4 , 5 ) δ generate this action group, the image of S 3 ֒ → S 6 . Con � B , { g 0 , g 1 }� ∼ = M 4 with congruences Con B 0 B α = | 012 | 345 | , β = | 03 | 14 | 25 | , γ = | 04 | 15 | 23 | , δ = | 05 | 13 | 24 | . L 9

  13. C ONSTRUCTION OF AN ALGEBRA A WITH Con A ∼ = L 9 . S TEP 1 Take a permutational algebra B = � B , F � with congruence lattice Con B ∼ = M 4 . Example: 1 B Let B = { 0 , 1 , . . . , 5 } index the elements of S 3 and consider the right regular action of S 3 on itself. γ α β g 0 = ( 0 , 4 )( 1 , 3 )( 2 , 5 ) and g 1 = ( 0 , 1 , 2 )( 3 , 4 , 5 ) δ generate this action group, the image of S 3 ֒ → S 6 . Con � B , { g 0 , g 1 }� ∼ = M 4 with congruences Con B 0 B α = | 012 | 345 | , β = | 03 | 14 | 25 | , γ = | 04 | 15 | 23 | , δ = | 05 | 13 | 24 | . Goal: expand B to an algebra A that has α “doubled” in Con A . α � α ∗ Con A

  14. C ONSTRUCTION OF AN ALGEBRA A WITH Con A ∼ = L 9 . S TEP 1 Take a permutational algebra B = � B , F � with congruence lattice Con B ∼ = M 4 . Example: 1 B Let B = { 0 , 1 , . . . , 5 } index the elements of S 3 and consider the right regular action of S 3 on itself. γ α β g 0 = ( 0 , 4 )( 1 , 3 )( 2 , 5 ) and g 1 = ( 0 , 1 , 2 )( 3 , 4 , 5 ) δ generate this action group, the image of S 3 ֒ → S 6 . Con � B , { g 0 , g 1 }� ∼ = M 4 with congruences Con B 0 B α = | 012 | 345 | , β = | 03 | 14 | 25 | , γ = | 04 | 15 | 23 | , δ = | 05 | 13 | 24 | . Goal: expand B to an algebra A that has α “doubled” in Con A . S TEP 2 Since α = Cg B ( 0 , 2 ) , we let A = B 0 ∪ B 1 ∪ B 2 where α � B 0 = { 0 , 1 , 2 , 3 , 4 , 5 } = B α ∗ B 1 = { 0 , 6 , 7 , 8 , 9 , 10 } Con A B 2 = { 11 , 12 , 2 , 13 , 14 , 15 } .

  15. C ONSTRUCTION OF AN ALGEBRA A WITH Con A ∼ = L 9 . S TEP 1 Take a permutational algebra B = � B , F � with congruence lattice Con B ∼ = M 4 . Example: 1 B Let B = { 0 , 1 , . . . , 5 } index the elements of S 3 and consider the right regular action of S 3 on itself. γ α β g 0 = ( 0 , 4 )( 1 , 3 )( 2 , 5 ) and g 1 = ( 0 , 1 , 2 )( 3 , 4 , 5 ) δ generate this action group, the image of S 3 ֒ → S 6 . Con � B , { g 0 , g 1 }� ∼ = M 4 with congruences Con B 0 B α = | 012 | 345 | , β = | 03 | 14 | 25 | , γ = | 04 | 15 | 23 | , δ = | 05 | 13 | 24 | . Goal: expand B to an algebra A that has α “doubled” in Con A . S TEP 2 Since α = Cg B ( 0 , 2 ) , we let A = B 0 ∪ B 1 ∪ B 2 where α � B 0 = { 0 , 1 , 2 , 3 , 4 , 5 } = B α ∗ B 1 = { 0 , 6 , 7 , 8 , 9 , 10 } Con A B 2 = { 11 , 12 , 2 , 13 , 14 , 15 } . S TEP 3 Define unary operations e 0 , e 1 , e 2 , s , g 0 e 0 , and g 1 e 0 .

  16. C ONSTRUCTION OF AN ALGEBRA A WITH Con A ∼ = L 9 . 1 A 1 B α � β ∗ γ ∗ δ ∗ γ α β δ α ∗ 0 A 0 B Con � A , F A � Con � B , { g 0 , g 1 }� α = | 0 , 1 , 2 , 6 , 7 , 11 , 12 | 3 , 4 , 5 | 8 , 9 , 10 , 13 , 14 , 15 | � α = | 0 , 1 , 2 | 3 , 4 , 5 | α ∗ = | 0 , 1 , 2 , 6 , 7 , 11 , 12 | 3 , 4 , 5 | 8 , 9 , 10 | 13 , 14 , 15 | β = | 0 , 3 | 1 , 4 | 2 , 5 | β ∗ = | 0 , 3 , 8 | 1 , 4 | 2 , 5 , 15 | 6 , 9 | 7 , 10 | 11 , 13 | 12 , 14 | γ = | 0 , 4 | 1 , 5 | 2 , 3 | γ ∗ = | 0 , 4 , 9 | 1 , 5 | 2 , 3 , 13 | 6 , 10 | 7 , 8 | 11 , 14 | 12 , 15 | δ = | 0 , 5 | 1 , 3 | 2 , 4 | δ ∗ = | 0 , 5 , 10 | 1 , 3 | 2 , 4 , 14 | 6 , 8 | 7 , 9 , 11 , 15 | 12 , 13 |

  17. C ONSTRUCTION OF AN ALGEBRA A WITH Con A ∼ = L 9 . 1 A 1 B α � β ∗ γ ∗ δ ∗ γ α β δ α ∗ 0 A 0 B Con � A , F A � Con � B , { g 0 , g 1 }� α = | 0 , 1 , 2 , 6 , 7 , 11 , 12 | 3 , 4 , 5 | 8 , 9 , 10 , 13 , 14 , 15 | � α = | 0 , 1 , 2 | 3 , 4 , 5 | α ∗ = | 0 , 1 , 2 , 6 , 7 , 11 , 12 | 3 , 4 , 5 | 8 , 9 , 10 | 13 , 14 , 15 | β = | 0 , 3 | 1 , 4 | 2 , 5 | β ∗ = | 0 , 3 , 8 | 1 , 4 | 2 , 5 , 15 | 6 , 9 | 7 , 10 | 11 , 13 | 12 , 14 | γ = | 0 , 4 | 1 , 5 | 2 , 3 | γ ∗ = | 0 , 4 , 9 | 1 , 5 | 2 , 3 , 13 | 6 , 10 | 7 , 8 | 11 , 14 | 12 , 15 | δ = | 0 , 5 | 1 , 3 | 2 , 4 | δ ∗ = | 0 , 5 , 10 | 1 , 3 | 2 , 4 , 14 | 6 , 8 | 7 , 9 , 11 , 15 | 12 , 13 | α = α ∗ ∩ B 2 = � β = β ∗ ∩ B 2 , α ∩ B 2 , . . .

  18. W HY DOES IT WORK ? Con � B , { g 0 , g 1 }� α = | 0 , 1 , 2 | 3 , 4 , 5 | 0 1 2 β = | 0 , 3 | 1 , 4 | 2 , 5 | 3 4 5 γ = | 0 , 4 | 1 , 5 | 2 , 3 | B 0 δ = | 0 , 5 | 1 , 3 | 2 , 4 |

  19. W HY DOES IT WORK ? Con � B , { g 0 , g 1 }� α = | 0 , 1 , 2 | 3 , 4 , 5 | 0 1 2 β = | 0 , 3 | 1 , 4 | 2 , 5 | 3 4 5 γ = | 0 , 4 | 1 , 5 | 2 , 3 | α B 0 δ = | 0 , 5 | 1 , 3 | 2 , 4 |

  20. W HY DOES IT WORK ? Con � B , { g 0 , g 1 }� α = | 0 , 1 , 2 | 3 , 4 , 5 | 0 1 2 β = | 0 , 3 | 1 , 4 | 2 , 5 | 3 4 5 γ = | 0 , 4 | 1 , 5 | 2 , 3 | β B 0 δ = | 0 , 5 | 1 , 3 | 2 , 4 |

  21. W HY DOES IT WORK ? Con � B , { g 0 , g 1 }� α = | 0 , 1 , 2 | 3 , 4 , 5 | 0 1 2 β = | 0 , 3 | 1 , 4 | 2 , 5 | 3 4 5 γ = | 0 , 4 | 1 , 5 | 2 , 3 | γ B 0 δ = | 0 , 5 | 1 , 3 | 2 , 4 |

  22. W HY DOES IT WORK ? Con � B , { g 0 , g 1 }� α = | 0 , 1 , 2 | 3 , 4 , 5 | 0 1 2 β = | 0 , 3 | 1 , 4 | 2 , 5 | 3 4 5 γ = | 0 , 4 | 1 , 5 | 2 , 3 | δ B 0 δ = | 0 , 5 | 1 , 3 | 2 , 4 |

  23. W HY DOES IT WORK ? Con � B , { g 0 , g 1 }� α = | 0 , 1 , 2 | 3 , 4 , 5 | 0 1 2 β = | 0 , 3 | 1 , 4 | 2 , 5 | 3 4 5 γ = | 0 , 4 | 1 , 5 | 2 , 3 | α B 0 δ = | 0 , 5 | 1 , 3 | 2 , 4 | B 0 = { 0 1 2 3 4 5 }

  24. W HY DOES IT WORK ? B 1 B 2 Con � B , { g 0 , g 1 }� 10 9 8 15 14 13 α = | 0 , 1 , 2 | 3 , 4 , 5 | 7 6 0 1 2 12 11 β = | 0 , 3 | 1 , 4 | 2 , 5 | 3 4 5 γ = | 0 , 4 | 1 , 5 | 2 , 3 | B 0 δ = | 0 , 5 | 1 , 3 | 2 , 4 | A = B 0 ∪ B 1 ∪ B 2 B 1 = { 0 6 7 8 9 10 } B 0 = { 0 1 2 3 4 5 } B 2 = { 11 12 2 13 14 15 }

  25. W HY DOES IT WORK ? B 1 B 2 Con � B , { g 0 , g 1 }� 10 9 8 15 14 13 α = | 0 , 1 , 2 | 3 , 4 , 5 | 7 6 0 1 2 12 11 β = | 0 , 3 | 1 , 4 | 2 , 5 | 3 4 5 γ = | 0 , 4 | 1 , 5 | 2 , 3 | B 0 δ = | 0 , 5 | 1 , 3 | 2 , 4 | A = B 0 ∪ B 1 ∪ B 2 Unary operations B 1 = { 0 6 7 8 9 10 } e 0 : A ։ B 0 e 1 : A ։ B 1 B 0 = { 0 1 2 3 4 5 } e 2 : A ։ B 2 s : A ։ B 0 B 2 = { 11 12 2 13 14 15 } e 0 g ge 0 : A ։ B 0 → B 0

  26. W HY DOES IT WORK ? B 2 Con � B , { g 0 , g 1 }� 10 9 8 15 14 13 7 α = | 0 , 1 , 2 | 3 , 4 , 5 | 6 0 1 2 12 11 β = | 0 , 3 | 1 , 4 | 2 , 5 | 3 4 5 γ = | 0 , 4 | 1 , 5 | 2 , 3 | B 0 δ = | 0 , 5 | 1 , 3 | 2 , 4 | A = B 0 ∪ B 1 ∪ B 2 Unary operations B 1 = { 0 6 7 8 9 10 } e 0 : A ։ B 0 e 1 : A ։ B 1 B 0 = { 0 1 2 3 4 5 } e 2 : A ։ B 2 s : A ։ B 0 B 2 = { 11 12 2 13 14 15 } e 0 g ge 0 : A ։ B 0 → B 0

  27. W HY DOES IT WORK ? B 2 Con � B , { g 0 , g 1 }� 10 9 7 15 14 13 8 6 α = | 0 , 1 , 2 | 3 , 4 , 5 | 0 1 2 12 11 β = | 0 , 3 | 1 , 4 | 2 , 5 | 3 4 5 γ = | 0 , 4 | 1 , 5 | 2 , 3 | B 0 δ = | 0 , 5 | 1 , 3 | 2 , 4 | A = B 0 ∪ B 1 ∪ B 2 Unary operations B 1 = { 0 6 7 8 9 10 } e 0 : A ։ B 0 e 1 : A ։ B 1 B 0 = { 0 1 2 3 4 5 } e 2 : A ։ B 2 s : A ։ B 0 B 2 = { 11 12 2 13 14 15 } e 0 g ge 0 : A ։ B 0 → B 0

  28. W HY DOES IT WORK ? 10 B 2 Con � B , { g 0 , g 1 }� 7 9 15 14 13 6 8 α = | 0 , 1 , 2 | 3 , 4 , 5 | 0 1 2 12 11 β = | 0 , 3 | 1 , 4 | 2 , 5 | 3 4 5 γ = | 0 , 4 | 1 , 5 | 2 , 3 | B 0 δ = | 0 , 5 | 1 , 3 | 2 , 4 | A = B 0 ∪ B 1 ∪ B 2 Unary operations B 1 = { 0 6 7 8 9 10 } e 0 : A ։ B 0 e 1 : A ։ B 1 B 0 = { 0 1 2 3 4 5 } e 2 : A ։ B 2 s : A ։ B 0 B 2 = { 11 12 2 13 14 15 } e 0 g ge 0 : A ։ B 0 → B 0

  29. W HY DOES IT WORK ? 10 B 2 Con � B , { g 0 , g 1 }� 7 9 15 14 13 6 8 α = | 0 , 1 , 2 | 3 , 4 , 5 | 0 1 2 12 11 β = | 0 , 3 | 1 , 4 | 2 , 5 | 3 4 5 γ = | 0 , 4 | 1 , 5 | 2 , 3 | B 0 δ = | 0 , 5 | 1 , 3 | 2 , 4 | A = B 0 ∪ B 1 ∪ B 2 Unary operations B 1 = { 0 6 7 8 9 10 } e 0 : A ։ B 0 e 1 : A ։ B 1 B 0 = { 0 1 2 3 4 5 } e 2 : A ։ B 2 s : A ։ B 0 B 2 = { 11 12 2 13 14 15 } e 0 g ge 0 : A ։ B 0 → B 0

  30. W HY DOES IT WORK ? 10 7 B 2 Con � B , { g 0 , g 1 }� 9 6 15 14 13 α = | 0 , 1 , 2 | 3 , 4 , 5 | 8 0 1 2 12 11 β = | 0 , 3 | 1 , 4 | 2 , 5 | 3 4 5 γ = | 0 , 4 | 1 , 5 | 2 , 3 | B 0 δ = | 0 , 5 | 1 , 3 | 2 , 4 | A = B 0 ∪ B 1 ∪ B 2 Unary operations B 1 = { 0 6 7 8 9 10 } e 0 : A ։ B 0 e 1 : A ։ B 1 B 0 = { 0 1 2 3 4 5 } e 2 : A ։ B 2 s : A ։ B 0 B 2 = { 11 12 2 13 14 15 } e 0 g ge 0 : A ։ B 0 → B 0

  31. W HY DOES IT WORK ? 7 10 B 2 Con � B , { g 0 , g 1 }� 6 9 15 14 13 α = | 0 , 1 , 2 | 3 , 4 , 5 | 0 8 1 2 12 11 β = | 0 , 3 | 1 , 4 | 2 , 5 | 3 4 5 γ = | 0 , 4 | 1 , 5 | 2 , 3 | B 0 δ = | 0 , 5 | 1 , 3 | 2 , 4 | A = B 0 ∪ B 1 ∪ B 2 Unary operations B 1 = { 0 6 7 8 9 10 } e 0 : A ։ B 0 e 1 : A ։ B 1 B 0 = { 0 1 2 3 4 5 } e 2 : A ։ B 2 s : A ։ B 0 B 2 = { 11 12 2 13 14 15 } e 0 g ge 0 : A ։ B 0 → B 0

  32. W HY DOES IT WORK ? 7 B 2 Con � B , { g 0 , g 1 }� 10 6 15 14 13 9 α = | 0 , 1 , 2 | 3 , 4 , 5 | 0 1 2 12 11 8 β = | 0 , 3 | 1 , 4 | 2 , 5 | 3 4 5 γ = | 0 , 4 | 1 , 5 | 2 , 3 | B 0 δ = | 0 , 5 | 1 , 3 | 2 , 4 | A = B 0 ∪ B 1 ∪ B 2 Unary operations B 1 = { 0 6 7 8 9 10 } e 0 : A ։ B 0 e 1 : A ։ B 1 B 0 = { 0 1 2 3 4 5 } e 2 : A ։ B 2 s : A ։ B 0 B 2 = { 11 12 2 13 14 15 } e 0 g ge 0 : A ։ B 0 → B 0

  33. W HY DOES IT WORK ? B 2 Con � B , { g 0 , g 1 }� 7 10 15 14 13 6 α = | 0 , 1 , 2 | 3 , 4 , 5 | 9 0 1 2 12 11 β = | 0 , 3 | 1 , 4 | 2 , 5 | 8 3 4 5 γ = | 0 , 4 | 1 , 5 | 2 , 3 | B 0 δ = | 0 , 5 | 1 , 3 | 2 , 4 | A = B 0 ∪ B 1 ∪ B 2 Unary operations B 1 = { 0 6 7 8 9 10 } e 0 : A ։ B 0 e 1 : A ։ B 1 B 0 = { 0 1 2 3 4 5 } e 2 : A ։ B 2 s : A ։ B 0 B 2 = { 11 12 2 13 14 15 } e 0 g ge 0 : A ։ B 0 → B 0

  34. W HY DOES IT WORK ? B 2 Con � B , { g 0 , g 1 }� 7 15 14 13 6 10 α = | 0 , 1 , 2 | 3 , 4 , 5 | 0 1 9 2 12 11 β = | 0 , 3 | 1 , 4 | 2 , 5 | 8 3 4 5 γ = | 0 , 4 | 1 , 5 | 2 , 3 | B 0 δ = | 0 , 5 | 1 , 3 | 2 , 4 | A = B 0 ∪ B 1 ∪ B 2 Unary operations B 1 = { 0 6 7 8 9 10 } e 0 : A ։ B 0 e 1 : A ։ B 1 B 0 = { 0 1 2 3 4 5 } e 2 : A ։ B 2 s : A ։ B 0 B 2 = { 11 12 2 13 14 15 } e 0 g ge 0 : A ։ B 0 → B 0

  35. W HY DOES IT WORK ? B 2 Con � B , { g 0 , g 1 }� 7 15 14 13 6 α = | 0 , 1 , 2 | 3 , 4 , 5 | 10 0 1 2 12 11 9 β = | 0 , 3 | 1 , 4 | 2 , 5 | 8 3 4 5 γ = | 0 , 4 | 1 , 5 | 2 , 3 | B 0 δ = | 0 , 5 | 1 , 3 | 2 , 4 | A = B 0 ∪ B 1 ∪ B 2 Unary operations B 1 = { 0 6 7 8 9 10 } e 0 : A ։ B 0 e 1 : A ։ B 1 B 0 = { 0 1 2 3 4 5 } e 2 : A ։ B 2 s : A ։ B 0 B 2 = { 11 12 2 13 14 15 } e 0 g ge 0 : A ։ B 0 → B 0

  36. W HY DOES IT WORK ? B 2 Con � B , { g 0 , g 1 }� 15 14 13 7 α = | 0 , 1 , 2 | 3 , 4 , 5 | 6 0 1 2 12 11 β = | 0 , 3 | 1 , 4 | 2 , 5 | 10 9 3 8 4 5 γ = | 0 , 4 | 1 , 5 | 2 , 3 | B 0 δ = | 0 , 5 | 1 , 3 | 2 , 4 | A = B 0 ∪ B 1 ∪ B 2 Unary operations B 1 = { 0 6 7 8 9 10 } e 0 : A ։ B 0 e 1 : A ։ B 1 B 0 = { 0 1 2 3 4 5 } e 2 : A ։ B 2 s : A ։ B 0 B 2 = { 11 12 2 13 14 15 } e 0 g ge 0 : A ։ B 0 → B 0

  37. W HY DOES IT WORK ? B 2 Con � B , { g 0 , g 1 }� 15 14 13 α = | 0 , 1 , 2 | 3 , 4 , 5 | 0 1 2 12 11 β = | 0 , 3 | 1 , 4 | 2 , 5 | 3 4 5 γ = | 0 , 4 | 1 , 5 | 2 , 3 | B 0 δ = | 0 , 5 | 1 , 3 | 2 , 4 | A = B 0 ∪ B 1 ∪ B 2 Unary operations B 1 = { 0 6 7 8 9 10 } e 0 : A ։ B 0 e 1 : A ։ B 1 B 0 = { 0 1 2 3 4 5 } e 2 : A ։ B 2 s : A ։ B 0 B 2 = { 11 12 2 13 14 15 } e 0 g ge 0 : A ։ B 0 → B 0

  38. W HY DOES IT WORK ? B 2 Con � B , { g 0 , g 1 }� 15 14 13 α = | 0 , 1 , 2 | 3 , 4 , 5 | 0 1 2 12 11 β = | 0 , 3 | 1 , 4 | 2 , 5 | 3 4 5 γ = | 0 , 4 | 1 , 5 | 2 , 3 | B 0 δ = | 0 , 5 | 1 , 3 | 2 , 4 | A = B 0 ∪ B 1 ∪ B 2 Unary operations B 1 = { 0 6 7 8 9 10 } e 0 : A ։ B 0 e 1 : A ։ B 1 B 0 = { 0 1 2 3 4 5 } e 2 : A ։ B 2 s : A ։ B 0 B 2 = { 11 12 2 13 14 15 } e 0 g ge 0 : A ։ B 0 → B 0

  39. W HY DOES IT WORK ? B 2 Con � B , { g 0 , g 1 }� 15 14 13 α = | 0 , 1 , 2 | 3 , 4 , 5 | 0 1 2 12 11 β = | 0 , 3 | 1 , 4 | 2 , 5 | 3 4 5 γ = | 0 , 4 | 1 , 5 | 2 , 3 | B 0 δ = | 0 , 5 | 1 , 3 | 2 , 4 | A = B 0 ∪ B 1 ∪ B 2 Unary operations B 1 = { 0 6 7 8 9 10 } e 0 : A ։ B 0 e 1 : A ։ B 1 B 0 = { 0 1 2 3 4 5 } e 2 : A ։ B 2 s : A ։ B 0 B 2 = { 11 12 2 13 14 15 } e 0 g ge 0 : A ։ B 0 → B 0

  40. W HY DOES IT WORK ? Con � B , { g 0 , g 1 }� 13 14 15 11 α = | 0 , 1 , 2 | 3 , 4 , 5 | 12 0 1 2 β = | 0 , 3 | 1 , 4 | 2 , 5 | 3 4 5 γ = | 0 , 4 | 1 , 5 | 2 , 3 | B 0 δ = | 0 , 5 | 1 , 3 | 2 , 4 | A = B 0 ∪ B 1 ∪ B 2 Unary operations B 1 = { 0 6 7 8 9 10 } e 0 : A ։ B 0 e 1 : A ։ B 1 B 0 = { 0 1 2 3 4 5 } e 2 : A ։ B 2 s : A ։ B 0 B 2 = { 11 12 2 13 14 15 } e 0 g ge 0 : A ։ B 0 → B 0

  41. W HY DOES IT WORK ? Con � B , { g 0 , g 1 }� 13 14 11 15 12 α = | 0 , 1 , 2 | 3 , 4 , 5 | 0 1 2 β = | 0 , 3 | 1 , 4 | 2 , 5 | 3 4 5 γ = | 0 , 4 | 1 , 5 | 2 , 3 | B 0 δ = | 0 , 5 | 1 , 3 | 2 , 4 | A = B 0 ∪ B 1 ∪ B 2 Unary operations B 1 = { 0 6 7 8 9 10 } e 0 : A ։ B 0 e 1 : A ։ B 1 B 0 = { 0 1 2 3 4 5 } e 2 : A ։ B 2 s : A ։ B 0 B 2 = { 11 12 2 13 14 15 } e 0 g ge 0 : A ։ B 0 → B 0

  42. W HY DOES IT WORK ? 13 Con � B , { g 0 , g 1 }� 14 11 15 12 α = | 0 , 1 , 2 | 3 , 4 , 5 | 0 1 2 β = | 0 , 3 | 1 , 4 | 2 , 5 | 3 4 5 γ = | 0 , 4 | 1 , 5 | 2 , 3 | B 0 δ = | 0 , 5 | 1 , 3 | 2 , 4 | A = B 0 ∪ B 1 ∪ B 2 Unary operations B 1 = { 0 6 7 8 9 10 } e 0 : A ։ B 0 e 1 : A ։ B 1 B 0 = { 0 1 2 3 4 5 } e 2 : A ։ B 2 s : A ։ B 0 B 2 = { 11 12 2 13 14 15 } e 0 g ge 0 : A ։ B 0 → B 0

  43. W HY DOES IT WORK ? 13 Con � B , { g 0 , g 1 }� 11 14 12 15 α = | 0 , 1 , 2 | 3 , 4 , 5 | 0 1 2 β = | 0 , 3 | 1 , 4 | 2 , 5 | 3 4 5 γ = | 0 , 4 | 1 , 5 | 2 , 3 | B 0 δ = | 0 , 5 | 1 , 3 | 2 , 4 | A = B 0 ∪ B 1 ∪ B 2 Unary operations B 1 = { 0 6 7 8 9 10 } e 0 : A ։ B 0 e 1 : A ։ B 1 B 0 = { 0 1 2 3 4 5 } e 2 : A ։ B 2 s : A ։ B 0 B 2 = { 11 12 2 13 14 15 } e 0 g ge 0 : A ։ B 0 → B 0

  44. W HY DOES IT WORK ? 13 11 Con � B , { g 0 , g 1 }� 14 12 α = | 0 , 1 , 2 | 3 , 4 , 5 | 15 0 1 2 β = | 0 , 3 | 1 , 4 | 2 , 5 | 3 4 5 γ = | 0 , 4 | 1 , 5 | 2 , 3 | B 0 δ = | 0 , 5 | 1 , 3 | 2 , 4 | A = B 0 ∪ B 1 ∪ B 2 Unary operations B 1 = { 0 6 7 8 9 10 } e 0 : A ։ B 0 e 1 : A ։ B 1 B 0 = { 0 1 2 3 4 5 } e 2 : A ։ B 2 s : A ։ B 0 B 2 = { 11 12 2 13 14 15 } e 0 g ge 0 : A ։ B 0 → B 0

  45. W HY DOES IT WORK ? 13 11 Con � B , { g 0 , g 1 }� 14 12 α = | 0 , 1 , 2 | 3 , 4 , 5 | 0 15 1 2 β = | 0 , 3 | 1 , 4 | 2 , 5 | 3 4 5 γ = | 0 , 4 | 1 , 5 | 2 , 3 | B 0 δ = | 0 , 5 | 1 , 3 | 2 , 4 | A = B 0 ∪ B 1 ∪ B 2 Unary operations B 1 = { 0 6 7 8 9 10 } e 0 : A ։ B 0 e 1 : A ։ B 1 B 0 = { 0 1 2 3 4 5 } e 2 : A ։ B 2 s : A ։ B 0 B 2 = { 11 12 2 13 14 15 } e 0 g ge 0 : A ։ B 0 → B 0

  46. W HY DOES IT WORK ? 11 Con � B , { g 0 , g 1 }� 13 12 14 α = | 0 , 1 , 2 | 3 , 4 , 5 | 0 1 2 15 β = | 0 , 3 | 1 , 4 | 2 , 5 | 3 4 5 γ = | 0 , 4 | 1 , 5 | 2 , 3 | B 0 δ = | 0 , 5 | 1 , 3 | 2 , 4 | A = B 0 ∪ B 1 ∪ B 2 Unary operations B 1 = { 0 6 7 8 9 10 } e 0 : A ։ B 0 e 1 : A ։ B 1 B 0 = { 0 1 2 3 4 5 } e 2 : A ։ B 2 s : A ։ B 0 B 2 = { 11 12 2 13 14 15 } e 0 g ge 0 : A ։ B 0 → B 0

  47. W HY DOES IT WORK ? Con � B , { g 0 , g 1 }� 11 13 12 α = | 0 , 1 , 2 | 3 , 4 , 5 | 14 0 1 2 β = | 0 , 3 | 1 , 4 | 2 , 5 | 15 3 4 5 γ = | 0 , 4 | 1 , 5 | 2 , 3 | B 0 δ = | 0 , 5 | 1 , 3 | 2 , 4 | A = B 0 ∪ B 1 ∪ B 2 Unary operations B 1 = { 0 6 7 8 9 10 } e 0 : A ։ B 0 e 1 : A ։ B 1 B 0 = { 0 1 2 3 4 5 } e 2 : A ։ B 2 s : A ։ B 0 B 2 = { 11 12 2 13 14 15 } e 0 g ge 0 : A ։ B 0 → B 0

  48. W HY DOES IT WORK ? Con � B , { g 0 , g 1 }� 11 13 12 α = | 0 , 1 , 2 | 3 , 4 , 5 | 0 14 1 2 β = | 0 , 3 | 1 , 4 | 2 , 5 | 15 3 4 5 γ = | 0 , 4 | 1 , 5 | 2 , 3 | B 0 δ = | 0 , 5 | 1 , 3 | 2 , 4 | A = B 0 ∪ B 1 ∪ B 2 Unary operations B 1 = { 0 6 7 8 9 10 } e 0 : A ։ B 0 e 1 : A ։ B 1 B 0 = { 0 1 2 3 4 5 } e 2 : A ։ B 2 s : A ։ B 0 B 2 = { 11 12 2 13 14 15 } e 0 g ge 0 : A ։ B 0 → B 0

  49. W HY DOES IT WORK ? Con � B , { g 0 , g 1 }� 11 12 α = | 0 , 1 , 2 | 3 , 4 , 5 | 13 0 1 2 14 β = | 0 , 3 | 1 , 4 | 2 , 5 | 15 3 4 5 γ = | 0 , 4 | 1 , 5 | 2 , 3 | B 0 δ = | 0 , 5 | 1 , 3 | 2 , 4 | A = B 0 ∪ B 1 ∪ B 2 Unary operations B 1 = { 0 6 7 8 9 10 } e 0 : A ։ B 0 e 1 : A ։ B 1 B 0 = { 0 1 2 3 4 5 } e 2 : A ։ B 2 s : A ։ B 0 B 2 = { 11 12 2 13 14 15 } e 0 g ge 0 : A ։ B 0 → B 0

  50. W HY DOES IT WORK ? Con � B , { g 0 , g 1 }� 11 α = | 0 , 1 , 2 | 3 , 4 , 5 | 12 0 1 2 β = | 0 , 3 | 1 , 4 | 2 , 5 | 13 14 3 4 15 5 γ = | 0 , 4 | 1 , 5 | 2 , 3 | B 0 δ = | 0 , 5 | 1 , 3 | 2 , 4 | A = B 0 ∪ B 1 ∪ B 2 Unary operations B 1 = { 0 6 7 8 9 10 } e 0 : A ։ B 0 e 1 : A ։ B 1 B 0 = { 0 1 2 3 4 5 } e 2 : A ։ B 2 s : A ։ B 0 B 2 = { 11 12 2 13 14 15 } e 0 g ge 0 : A ։ B 0 → B 0

  51. W HY DOES IT WORK ? Con � B , { g 0 , g 1 }� α = | 0 , 1 , 2 | 3 , 4 , 5 | 0 1 2 β = | 0 , 3 | 1 , 4 | 2 , 5 | 3 4 5 γ = | 0 , 4 | 1 , 5 | 2 , 3 | B 0 δ = | 0 , 5 | 1 , 3 | 2 , 4 | A = B 0 ∪ B 1 ∪ B 2 Unary operations B 1 = { 0 6 7 8 9 10 } e 0 : A ։ B 0 e 1 : A ։ B 1 B 0 = { 0 1 2 3 4 5 } e 2 : A ։ B 2 s : A ։ B 0 B 2 = { 11 12 2 13 14 15 } e 0 g ge 0 : A ։ B 0 → B 0

  52. W HY DOES IT WORK ? B 1 B 2 Con � B , { g 0 , g 1 }� 10 9 8 15 14 13 α = | 0 , 1 , 2 | 3 , 4 , 5 | 7 6 0 1 2 12 11 β = | 0 , 3 | 1 , 4 | 2 , 5 | 3 4 5 γ = | 0 , 4 | 1 , 5 | 2 , 3 | B 0 δ = | 0 , 5 | 1 , 3 | 2 , 4 | A = B 0 ∪ B 1 ∪ B 2 Unary operations B 1 = { 0 6 7 8 9 10 } e 0 : A ։ B 0 e 1 : A ։ B 1 B 0 = { 0 1 2 3 4 5 } e 2 : A ։ B 2 s : A ։ B 0 B 2 = { 11 12 2 13 14 15 } e 0 g ge 0 : A ։ B 0 → B 0

  53. W HY DOES IT WORK ? B 1 Con � B , { g 0 , g 1 }� 13 14 10 9 8 15 11 α = | 0 , 1 , 2 | 3 , 4 , 5 | 12 7 6 0 1 2 β = | 0 , 3 | 1 , 4 | 2 , 5 | 3 4 5 γ = | 0 , 4 | 1 , 5 | 2 , 3 | B 0 δ = | 0 , 5 | 1 , 3 | 2 , 4 | A = B 0 ∪ B 1 ∪ B 2 Unary operations B 1 = { 0 6 7 8 9 10 } e 0 : A ։ B 0 e 1 : A ։ B 1 B 0 = { 0 1 2 3 4 5 } e 2 : A ։ B 2 s : A ։ B 0 B 2 = { 11 12 2 13 14 15 } e 0 g ge 0 : A ։ B 0 → B 0

  54. W HY DOES IT WORK ? B 1 Con � B , { g 0 , g 1 }� 13 14 10 9 8 11 15 12 α = | 0 , 1 , 2 | 3 , 4 , 5 | 7 6 0 1 2 β = | 0 , 3 | 1 , 4 | 2 , 5 | 3 4 5 γ = | 0 , 4 | 1 , 5 | 2 , 3 | B 0 δ = | 0 , 5 | 1 , 3 | 2 , 4 | A = B 0 ∪ B 1 ∪ B 2 Unary operations B 1 = { 0 6 7 8 9 10 } e 0 : A ։ B 0 e 1 : A ։ B 1 B 0 = { 0 1 2 3 4 5 } e 2 : A ։ B 2 s : A ։ B 0 B 2 = { 11 12 2 13 14 15 } e 0 g ge 0 : A ։ B 0 → B 0

  55. W HY DOES IT WORK ? 13 B 1 Con � B , { g 0 , g 1 }� 14 11 10 9 8 15 12 α = | 0 , 1 , 2 | 3 , 4 , 5 | 7 6 0 1 2 β = | 0 , 3 | 1 , 4 | 2 , 5 | 3 4 5 γ = | 0 , 4 | 1 , 5 | 2 , 3 | B 0 δ = | 0 , 5 | 1 , 3 | 2 , 4 | A = B 0 ∪ B 1 ∪ B 2 Unary operations B 1 = { 0 6 7 8 9 10 } e 0 : A ։ B 0 e 1 : A ։ B 1 B 0 = { 0 1 2 3 4 5 } e 2 : A ։ B 2 s : A ։ B 0 B 2 = { 11 12 2 13 14 15 } e 0 g ge 0 : A ։ B 0 → B 0

  56. W HY DOES IT WORK ? 13 B 1 Con � B , { g 0 , g 1 }� 11 14 10 9 8 12 15 α = | 0 , 1 , 2 | 3 , 4 , 5 | 7 6 0 1 2 β = | 0 , 3 | 1 , 4 | 2 , 5 | 3 4 5 γ = | 0 , 4 | 1 , 5 | 2 , 3 | B 0 δ = | 0 , 5 | 1 , 3 | 2 , 4 | A = B 0 ∪ B 1 ∪ B 2 Unary operations B 1 = { 0 6 7 8 9 10 } e 0 : A ։ B 0 e 1 : A ։ B 1 B 0 = { 0 1 2 3 4 5 } e 2 : A ։ B 2 s : A ։ B 0 B 2 = { 11 12 2 13 14 15 } e 0 g ge 0 : A ։ B 0 → B 0

  57. W HY DOES IT WORK ? 13 B 1 11 Con � B , { g 0 , g 1 }� 14 10 9 8 12 α = | 0 , 1 , 2 | 3 , 4 , 5 | 15 7 6 0 1 2 β = | 0 , 3 | 1 , 4 | 2 , 5 | 3 4 5 γ = | 0 , 4 | 1 , 5 | 2 , 3 | B 0 δ = | 0 , 5 | 1 , 3 | 2 , 4 | A = B 0 ∪ B 1 ∪ B 2 Unary operations B 1 = { 0 6 7 8 9 10 } e 0 : A ։ B 0 e 1 : A ։ B 1 B 0 = { 0 1 2 3 4 5 } e 2 : A ։ B 2 s : A ։ B 0 B 2 = { 11 12 2 13 14 15 } e 0 g ge 0 : A ։ B 0 → B 0

  58. W HY DOES IT WORK ? 13 11 B 1 Con � B , { g 0 , g 1 }� 10 9 8 14 12 α = | 0 , 1 , 2 | 3 , 4 , 5 | 7 6 0 15 1 2 β = | 0 , 3 | 1 , 4 | 2 , 5 | 3 4 5 γ = | 0 , 4 | 1 , 5 | 2 , 3 | B 0 δ = | 0 , 5 | 1 , 3 | 2 , 4 | A = B 0 ∪ B 1 ∪ B 2 Unary operations B 1 = { 0 6 7 8 9 10 } e 0 : A ։ B 0 e 1 : A ։ B 1 B 0 = { 0 1 2 3 4 5 } e 2 : A ։ B 2 s : A ։ B 0 B 2 = { 11 12 2 13 14 15 } e 0 g ge 0 : A ։ B 0 → B 0

  59. W HY DOES IT WORK ? B 1 11 Con � B , { g 0 , g 1 }� 13 10 9 8 12 14 α = | 0 , 1 , 2 | 3 , 4 , 5 | 7 6 0 1 2 15 β = | 0 , 3 | 1 , 4 | 2 , 5 | 3 4 5 γ = | 0 , 4 | 1 , 5 | 2 , 3 | B 0 δ = | 0 , 5 | 1 , 3 | 2 , 4 | A = B 0 ∪ B 1 ∪ B 2 Unary operations B 1 = { 0 6 7 8 9 10 } e 0 : A ։ B 0 e 1 : A ։ B 1 B 0 = { 0 1 2 3 4 5 } e 2 : A ։ B 2 s : A ։ B 0 B 2 = { 11 12 2 13 14 15 } e 0 g ge 0 : A ։ B 0 → B 0

  60. W HY DOES IT WORK ? B 1 Con � B , { g 0 , g 1 }� 11 13 10 9 8 12 α = | 0 , 1 , 2 | 3 , 4 , 5 | 14 7 6 0 1 2 β = | 0 , 3 | 1 , 4 | 2 , 5 | 15 3 4 5 γ = | 0 , 4 | 1 , 5 | 2 , 3 | B 0 δ = | 0 , 5 | 1 , 3 | 2 , 4 | A = B 0 ∪ B 1 ∪ B 2 Unary operations B 1 = { 0 6 7 8 9 10 } e 0 : A ։ B 0 e 1 : A ։ B 1 B 0 = { 0 1 2 3 4 5 } e 2 : A ։ B 2 s : A ։ B 0 B 2 = { 11 12 2 13 14 15 } e 0 g ge 0 : A ։ B 0 → B 0

  61. W HY DOES IT WORK ? B 1 Con � B , { g 0 , g 1 }� 11 10 9 8 13 12 α = | 0 , 1 , 2 | 3 , 4 , 5 | 7 6 0 14 1 2 β = | 0 , 3 | 1 , 4 | 2 , 5 | 15 3 4 5 γ = | 0 , 4 | 1 , 5 | 2 , 3 | B 0 δ = | 0 , 5 | 1 , 3 | 2 , 4 | A = B 0 ∪ B 1 ∪ B 2 Unary operations B 1 = { 0 6 7 8 9 10 } e 0 : A ։ B 0 e 1 : A ։ B 1 B 0 = { 0 1 2 3 4 5 } e 2 : A ։ B 2 s : A ։ B 0 B 2 = { 11 12 2 13 14 15 } e 0 g ge 0 : A ։ B 0 → B 0

  62. W HY DOES IT WORK ? B 1 Con � B , { g 0 , g 1 }� 10 9 8 11 12 α = | 0 , 1 , 2 | 3 , 4 , 5 | 13 7 6 0 1 2 14 β = | 0 , 3 | 1 , 4 | 2 , 5 | 15 3 4 5 γ = | 0 , 4 | 1 , 5 | 2 , 3 | B 0 δ = | 0 , 5 | 1 , 3 | 2 , 4 | A = B 0 ∪ B 1 ∪ B 2 Unary operations B 1 = { 0 6 7 8 9 10 } e 0 : A ։ B 0 e 1 : A ։ B 1 B 0 = { 0 1 2 3 4 5 } e 2 : A ։ B 2 s : A ։ B 0 B 2 = { 11 12 2 13 14 15 } e 0 g ge 0 : A ։ B 0 → B 0

  63. W HY DOES IT WORK ? B 1 Con � B , { g 0 , g 1 }� 10 9 8 11 α = | 0 , 1 , 2 | 3 , 4 , 5 | 12 7 6 0 1 2 β = | 0 , 3 | 1 , 4 | 2 , 5 | 13 14 3 4 15 5 γ = | 0 , 4 | 1 , 5 | 2 , 3 | B 0 δ = | 0 , 5 | 1 , 3 | 2 , 4 | A = B 0 ∪ B 1 ∪ B 2 Unary operations B 1 = { 0 6 7 8 9 10 } e 0 : A ։ B 0 e 1 : A ։ B 1 B 0 = { 0 1 2 3 4 5 } e 2 : A ։ B 2 s : A ։ B 0 B 2 = { 11 12 2 13 14 15 } e 0 g ge 0 : A ։ B 0 → B 0

  64. W HY DOES IT WORK ? B 1 Con � B , { g 0 , g 1 }� 10 9 8 α = | 0 , 1 , 2 | 3 , 4 , 5 | 7 6 0 1 2 β = | 0 , 3 | 1 , 4 | 2 , 5 | 3 4 5 γ = | 0 , 4 | 1 , 5 | 2 , 3 | B 0 δ = | 0 , 5 | 1 , 3 | 2 , 4 | A = B 0 ∪ B 1 ∪ B 2 Unary operations B 1 = { 0 6 7 8 9 10 } e 0 : A ։ B 0 e 1 : A ։ B 1 B 0 = { 0 1 2 3 4 5 } e 2 : A ։ B 2 s : A ։ B 0 B 2 = { 11 12 2 13 14 15 } e 0 g ge 0 : A ։ B 0 → B 0

  65. W HY DOES IT WORK ? B 1 Con � B , { g 0 , g 1 }� 10 9 8 α = | 0 , 1 , 2 | 3 , 4 , 5 | 7 6 0 1 2 β = | 0 , 3 | 1 , 4 | 2 , 5 | 3 4 5 γ = | 0 , 4 | 1 , 5 | 2 , 3 | B 0 δ = | 0 , 5 | 1 , 3 | 2 , 4 | A = B 0 ∪ B 1 ∪ B 2 Unary operations B 1 = { 0 6 7 8 9 10 } e 0 : A ։ B 0 e 1 : A ։ B 1 B 0 = { 0 1 2 3 4 5 } e 2 : A ։ B 2 s : A ։ B 0 B 2 = { 11 12 2 13 14 15 } e 0 g ge 0 : A ։ B 0 → B 0

  66. W HY DOES IT WORK ? B 1 Con � B , { g 0 , g 1 }� 10 9 8 α = | 0 , 1 , 2 | 3 , 4 , 5 | 7 6 0 1 2 β = | 0 , 3 | 1 , 4 | 2 , 5 | 3 4 5 γ = | 0 , 4 | 1 , 5 | 2 , 3 | δ = | 0 , 5 | 1 , 3 | 2 , 4 | A = B 0 ∪ B 1 ∪ B 2 Unary operations B 1 = { 0 6 7 8 9 10 } e 0 : A ։ B 0 e 1 : A ։ B 1 B 0 = { 0 1 2 3 4 5 } e 2 : A ։ B 2 s : A ։ B 0 B 2 = { 11 12 2 13 14 15 } e 0 g ge 0 : A ։ B 0 → B 0

  67. W HY DOES IT WORK ? B 1 Con � B , { g 0 , g 1 }� 10 9 8 2 α = | 0 , 1 , 2 | 3 , 4 , 5 | 1 7 6 0 β = | 0 , 3 | 1 , 4 | 2 , 5 | 5 4 3 γ = | 0 , 4 | 1 , 5 | 2 , 3 | δ = | 0 , 5 | 1 , 3 | 2 , 4 | A = B 0 ∪ B 1 ∪ B 2 Unary operations B 1 = { 0 6 7 8 9 10 } e 0 : A ։ B 0 e 1 : A ։ B 1 B 0 = { 0 1 2 3 4 5 } e 2 : A ։ B 2 s : A ։ B 0 B 2 = { 11 12 2 13 14 15 } e 0 g ge 0 : A ։ B 0 → B 0

  68. W HY DOES IT WORK ? B 1 Con � B , { g 0 , g 1 }� 10 9 8 2 1 α = | 0 , 1 , 2 | 3 , 4 , 5 | 5 7 6 0 4 β = | 0 , 3 | 1 , 4 | 2 , 5 | 3 γ = | 0 , 4 | 1 , 5 | 2 , 3 | δ = | 0 , 5 | 1 , 3 | 2 , 4 | A = B 0 ∪ B 1 ∪ B 2 Unary operations B 1 = { 0 6 7 8 9 10 } e 0 : A ։ B 0 e 1 : A ։ B 1 B 0 = { 0 1 2 3 4 5 } e 2 : A ։ B 2 s : A ։ B 0 B 2 = { 11 12 2 13 14 15 } e 0 g ge 0 : A ։ B 0 → B 0

  69. W HY DOES IT WORK ? B 1 Con � B , { g 0 , g 1 }� 2 10 9 8 1 5 α = | 0 , 1 , 2 | 3 , 4 , 5 | 7 6 0 4 β = | 0 , 3 | 1 , 4 | 2 , 5 | 3 γ = | 0 , 4 | 1 , 5 | 2 , 3 | δ = | 0 , 5 | 1 , 3 | 2 , 4 | A = B 0 ∪ B 1 ∪ B 2 Unary operations B 1 = { 0 6 7 8 9 10 } e 0 : A ։ B 0 e 1 : A ։ B 1 B 0 = { 0 1 2 3 4 5 } e 2 : A ։ B 2 s : A ։ B 0 B 2 = { 11 12 2 13 14 15 } e 0 g ge 0 : A ։ B 0 → B 0

  70. W HY DOES IT WORK ? B 1 Con � B , { g 0 , g 1 }� 2 5 10 9 8 1 α = | 0 , 1 , 2 | 3 , 4 , 5 | 4 7 6 0 β = | 0 , 3 | 1 , 4 | 2 , 5 | 3 γ = | 0 , 4 | 1 , 5 | 2 , 3 | δ = | 0 , 5 | 1 , 3 | 2 , 4 | A = B 0 ∪ B 1 ∪ B 2 Unary operations B 1 = { 0 6 7 8 9 10 } e 0 : A ։ B 0 e 1 : A ։ B 1 B 0 = { 0 1 2 3 4 5 } e 2 : A ։ B 2 s : A ։ B 0 B 2 = { 11 12 2 13 14 15 } e 0 g ge 0 : A ։ B 0 → B 0

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