h ow to find a finite algebra with a given congruence
play

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

Oct 16, 2023 •278 likes •1.67k views

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

Download Presentation
Download Policy: The content available on the website is offered to you 'AS IS' for your personal information and use only. It cannot be commercialized, licensed, or distributed on other websites without prior consent from the author. To download a presentation, simply click this link. If you encounter any difficulties during the download process, it's possible that the publisher has removed the file from their server.

Recommend

Quick course in Universal Algebra and Tame Congruence Theory Ross Willard University of

Quick course in Universal Algebra and Tame Congruence Theory Ross Willard University of

Quick course in Universal Algebra and Tame Congruence Theory Ross Willard University of Waterloo, Canada Workshop on Universal Algebra and the Constraint Satisfaction Problem Nashville, June 2007 (with revisions added after the presentation)

802 views • 35 slides
WHY? rem(a,n) = rem(b,n) 66666663 example: 30 12 (mod 9) 788253 - since xxxxxxx0

WHY? rem(a,n) = rem(b,n) 66666663 example: 30 12 (mod 9) 788253 - since xxxxxxx0

Congruence mod n Mathematics for Computer Science MIT 6.042J/18.062J Def : a b (mod n) Congruences: iff n|(a - - b) example: 30 12 (mod 9) arithmetic (mod n) since 9 divides (30 - 12) congruence.1 congruence.2 Albert R Meyer, March

417 views • 6 slides
The local multiplier algebra of a C -algebra with finite-dimensional irreducible

The local multiplier algebra of a C -algebra with finite-dimensional irreducible

The local multiplier algebra of a C -algebra with finite-dimensional irreducible representations Ilja Gogi c Department of Mathematics, University of Zagreb (Croatia) and Department of Mathematics and Informatics, University of Novi Sad

785 views • 53 slides
Constraint Satisfaction Problems A Survey Ross Willard University of Waterloo, CAN Algebra

Constraint Satisfaction Problems A Survey Ross Willard University of Waterloo, CAN Algebra

Constraint Satisfaction Problems A Survey Ross Willard University of Waterloo, CAN Algebra & Algorithms University of Colorado May 19, 2016 (with corrections) CSP = specifications of subpowers of a finite algebra Fix a finite algebra A .

578 views • 18 slides
Appendix B Matrices and Matrix Algebra The interested readers may find a complete treatment in

Appendix B Matrices and Matrix Algebra The interested readers may find a complete treatment in

Appendix B Matrices and Matrix Algebra The interested readers may find a complete treatment in several Italian [33, 40] or English textbooks; a classical one is [15], while for linear algebra my preference goes to the book of Strang [45]. His

398 views • 37 slides
Checking NFA Equivalence with Bisimulations up to Congruence Filippo Bonchi and Damien Pous

Checking NFA Equivalence with Bisimulations up to Congruence Filippo Bonchi and Damien Pous

Checking NFA Equivalence with Bisimulations up to Congruence Filippo Bonchi and Damien Pous CNRS, LIP, ENS Lyon POPL, Roma, 25.1.2o13 Language equivalence of finite automata Useful for model checking: check that a program refines its

1.36k views • 92 slides
Sobriety and congruence biframes Graham Manuell graham@manuell.me University of Edinburgh

Sobriety and congruence biframes Graham Manuell graham@manuell.me University of Edinburgh

Sobriety and congruence biframes Graham Manuell graham@manuell.me University of Edinburgh Workshop on Algebra, Logic and Topology September 2018 1 Overview Sober spaces are the only topological spaces that can be faithfully represented

570 views • 36 slides
Congruence in univalent type theory Luis Scoccola lscoccol@uwo.ca University of Western Ontario

Congruence in univalent type theory Luis Scoccola lscoccol@uwo.ca University of Western Ontario

Congruence in univalent type theory Luis Scoccola lscoccol@uwo.ca University of Western Ontario June 11, 2019 Goals for the talk Congruence and congruence closure for propositional equality. Solution of Selsam & de Moura for a

940 views • 14 slides
Congruence permutable Fregean varieties Katarzyna Somczyska Pedagogical University in Krakw

Congruence permutable Fregean varieties Katarzyna Somczyska Pedagogical University in Krakw

Outline Introduction The structure of finite algebras Free algebras Congruence permutable Fregean varieties Katarzyna Somczyska Pedagogical University in Krakw Katarzyna Somczyska Congruence permutable Fregean varieties Outline

516 views • 47 slides
Constraints, Graphs, Algebra, Logic, and complexity Moshe Y. Vardi Rice University Constraint

Constraints, Graphs, Algebra, Logic, and complexity Moshe Y. Vardi Rice University Constraint

Constraints, Graphs, Algebra, Logic, and complexity Moshe Y. Vardi Rice University Constraint Satisfaction Problem (CSP) Input: ( V, D, C ) : A finite set V of variables A finite set D of values A finite set C of constraints

532 views • 37 slides
The Finite Embeddability Property for Topological Quasi-Boolean Algebra 5 Zhe Lin 1 and Mihir

The Finite Embeddability Property for Topological Quasi-Boolean Algebra 5 Zhe Lin 1 and Mihir

The Finite Embeddability Property for Topological Quasi-Boolean Algebra 5 Zhe Lin 1 and Mihir Kumar Chakraborty 2 Southwest University, Chongqing, China. 1 School of Cognitive Science, Jadavpur University, India 2 2019/03/03 Zhe Lin1 and Mihir

594 views • 27 slides
Algorithms & Techniques for Dense Linear Algebra over Small Finite Fields Martin R. Albrecht

Algorithms & Techniques for Dense Linear Algebra over Small Finite Fields Martin R. Albrecht

Algorithms & Techniques for Dense Linear Algebra over Small Finite Fields Martin R. Albrecht (martinralbrecht+summerschool@googlemail.com) POLSYS Team, UPMC, Paris, France ECrypt II PhD Summer School Outline F 2 Gray Codes Multiplication

1.01k views • 75 slides
Integral D-finite Functions Manuel Kauers Institute for Algebra Johannes Kepler University Linz,

Integral D-finite Functions Manuel Kauers Institute for Algebra Johannes Kepler University Linz,

Integral D-finite Functions Manuel Kauers Institute for Algebra Johannes Kepler University Linz, Austria joint work with Christoph Koutschan 1 Integral D-finite Functions p 0 ( x ) f ( x ) + p 1 ( x ) f ( x ) + + p r ( x ) f ( r

968 views • 84 slides
Automata, Logic and Algebra for (Finite) Word Transductions Emmanuel Filiot Universit e libre

Automata, Logic and Algebra for (Finite) Word Transductions Emmanuel Filiot Universit e libre

Introduction Transducers Logic Algebra New logic Summary Automata, Logic and Algebra for (Finite) Word Transductions Emmanuel Filiot Universit e libre de Bruxelles & FNRS ACTS 2017, Chennai 1 / 31 Introduction Transducers Logic

1.02k views • 90 slides
Approximate Congruence Detection of Model Features for Reverse Engineering C. H. Gao, F. C.

Approximate Congruence Detection of Model Features for Reverse Engineering C. H. Gao, F. C.

Approximate Congruence Detection of Model Features for Reverse Engineering C. H. Gao, F. C. Langbein, A. D. Marshall and R. R. Martin ralph@cs.cf.ac.uk Department of Computer Science, Cardiff University, Wales, UK Congruence Detection

388 views • 28 slides
Finite groups with some CEP -subgroups Izabela Agata Malinowska Institute of Mathematics

Finite groups with some CEP -subgroups Izabela Agata Malinowska Institute of Mathematics

Finite groups with some CEP -subgroups Izabela Agata Malinowska Institute of Mathematics University of Biaystok, Poland Warsaw, 19-22.06.2014 Izabela Agata Malinowska Finite groups with some CEP-subgroups Congruence Extension Property All

993 views • 54 slides
Influence of Beckmann, McGuire, and Winsten's Studies in the Economics of Transportation on

Influence of Beckmann, McGuire, and Winsten's Studies in the Economics of Transportation on

Influence of Beckmann, McGuire, and Winsten's Studies in the Economics of Transportation on Innovations in Modeling, Methodological Developments, and Applications Anna Nagurney Department of Finance and Operations Management Isenberg School

1.2k views • 97 slides
Role of FFR-Guided PCI in Patients with Stable CAD William F. Fearon, MD Professor of Medicine

Role of FFR-Guided PCI in Patients with Stable CAD William F. Fearon, MD Professor of Medicine

12/8/19 Role of FFR-Guided PCI in Patients with Stable CAD William F. Fearon, MD Professor of Medicine Director, Interventional Cardiology Stanford University School of Medicine 1 Conflicts of Interest n Research grants from Medtronic and

463 views • 30 slides
Engaging Community: Civic Engagement in Higher Education Marisa Hightower Associate Director

Engaging Community: Civic Engagement in Higher Education Marisa Hightower Associate Director

Engaging Community: Civic Engagement in Higher Education Marisa Hightower Associate Director Florida Civic Advance 2017 Summit, November 4, 2017 The Engaged University Fostering Interdisciplinarity Encouraging Educating for Collaboration

217 views • 6 slides
Example Sentences and Making them Useful for Theoretical and Computational Linguistics Stefan M

Example Sentences and Making them Useful for Theoretical and Computational Linguistics Stefan M

Example Sentences and Making them Useful for Theoretical and Computational Linguistics Stefan M uller Email: Stefan.Mueller@cl.uni-bremen.de http://www.cl.uni-bremen.de/stefan/ DGfS-Jahrestagung Mainz, 27.02.2004 Outline Why test

774 views • 40 slides
Dark Matter Freeze-in and LHC displaced signatures Alberto Mariotti HEP@ Based on

Dark Matter Freeze-in and LHC displaced signatures Alberto Mariotti HEP@ Based on

Dark Matter Freeze-in and LHC displaced signatures Alberto Mariotti HEP@ Based on arXiv:1805.04423 with: Lorenzo Calibbi, Laura Lopez Honorez, Steven Lowette GGI workhop - Beyond Standard Model: Where do we go from here? 27 August 2018 Plan

802 views • 34 slides
Lectures and Exercises Lectures and Exercises Lectures Lectures Researcher Stein L.

Lectures and Exercises Lectures and Exercises Lectures Lectures Researcher Stein L.

1 TDT4215 Web-intelligence TDT4215 Web intelligence Main topics: Information Retrieval Information Retrieval Large textual document collections Text mining NLP for document analysis NLP for document analysis

495 views • 16 slides
Justine Schneider Researcher Tanya Myers Writer, Director 2 Challenging care: role of HCAs in

Justine Schneider Researcher Tanya Myers Writer, Director 2 Challenging care: role of HCAs in

Justine Schneider Researcher Tanya Myers Writer, Director 2 Challenging care: role of HCAs in dementia wards Joanne Lloyd, Kezia Scales, Simon Bailey 3 Observations Observation Reflection Themes The last I know I Attachment, patient we

769 views • 29 slides
Faculty Disclosure New and Improved Choices for Endografts: Consultant in AAA field: WLGore,

Faculty Disclosure New and Improved Choices for Endografts: Consultant in AAA field: WLGore,

4/19/2013 Faculty Disclosure New and Improved Choices for Endografts: Consultant in AAA field: WLGore, Medtronic, Cordis Matching Anatomy to Device Grant/research support in AAA field from Cook, WLGore, Cordis, Medtronic, Bolton,

158 views • 15 slides

More recommend