mathematics

Type Members

1. trait Field [T] extends Ring[T]

2. trait GeneratedGroupImplicits extends AnyRef

3. trait GeneratedMonoidImplicits extends AnyRef

4. trait GeneratedRingImplicits extends AnyRef

5. trait Group [T] extends Monoid[T]

6. class ListMonoid [T] extends Monoid[List[T]]

   List concatenation monoid.

7. class MapGroup [K, V] extends MapMonoid[K, V] with Group[Map[K, V]]

   You can think of this as a Sparse vector group

8. class MapMonoid [K, V] extends Monoid[Map[K, V]]

   You can think of this as a Sparse vector monoid

9. class MapRing [K, V] extends MapGroup[K, V] with Ring[Map[K, V]]

   You can think of this as a Sparse vector ring

10. trait Monoid [T] extends Serializable

    Monoid (take a deep breath, and relax about the weird name): This is a class that has an additive identify (called zero), and plus method that is associative: a+(b+c) = (a+b)+c and a+0=a, 0+a=a

11. trait Ring [T] extends Group[T]

12. class SetMonoid [T] extends Monoid[Set[T]]

    Set union monoid.

13. class Tuple10Group [A, B, C, D, E, F, G, H, I, J] extends Group[(A, B, C, D, E, F, G, H, I, J)]

    Combine 10 groups into a product group

14. class Tuple10Monoid [A, B, C, D, E, F, G, H, I, J] extends Monoid[(A, B, C, D, E, F, G, H, I, J)]

    Combine 10 monoids into a product monoid

15. class Tuple10Ring [A, B, C, D, E, F, G, H, I, J] extends Ring[(A, B, C, D, E, F, G, H, I, J)]

    Combine 10 rings into a product ring

16. class Tuple11Group [A, B, C, D, E, F, G, H, I, J, K] extends Group[(A, B, C, D, E, F, G, H, I, J, K)]

    Combine 11 groups into a product group

17. class Tuple11Monoid [A, B, C, D, E, F, G, H, I, J, K] extends Monoid[(A, B, C, D, E, F, G, H, I, J, K)]

    Combine 11 monoids into a product monoid

18. class Tuple11Ring [A, B, C, D, E, F, G, H, I, J, K] extends Ring[(A, B, C, D, E, F, G, H, I, J, K)]

    Combine 11 rings into a product ring

19. class Tuple12Group [A, B, C, D, E, F, G, H, I, J, K, L] extends Group[(A, B, C, D, E, F, G, H, I, J, K, L)]

    Combine 12 groups into a product group

20. class Tuple12Monoid [A, B, C, D, E, F, G, H, I, J, K, L] extends Monoid[(A, B, C, D, E, F, G, H, I, J, K, L)]

    Combine 12 monoids into a product monoid

21. class Tuple12Ring [A, B, C, D, E, F, G, H, I, J, K, L] extends Ring[(A, B, C, D, E, F, G, H, I, J, K, L)]

    Combine 12 rings into a product ring

22. class Tuple13Group [A, B, C, D, E, F, G, H, I, J, K, L, M] extends Group[(A, B, C, D, E, F, G, H, I, J, K, L, M)]

    Combine 13 groups into a product group

23. class Tuple13Monoid [A, B, C, D, E, F, G, H, I, J, K, L, M] extends Monoid[(A, B, C, D, E, F, G, H, I, J, K, L, M)]

    Combine 13 monoids into a product monoid

24. class Tuple13Ring [A, B, C, D, E, F, G, H, I, J, K, L, M] extends Ring[(A, B, C, D, E, F, G, H, I, J, K, L, M)]

    Combine 13 rings into a product ring

25. class Tuple14Group [A, B, C, D, E, F, G, H, I, J, K, L, M, N] extends Group[(A, B, C, D, E, F, G, H, I, J, K, L, M, N)]

    Combine 14 groups into a product group

26. class Tuple14Monoid [A, B, C, D, E, F, G, H, I, J, K, L, M, N] extends Monoid[(A, B, C, D, E, F, G, H, I, J, K, L, M, N)]

    Combine 14 monoids into a product monoid

27. class Tuple14Ring [A, B, C, D, E, F, G, H, I, J, K, L, M, N] extends Ring[(A, B, C, D, E, F, G, H, I, J, K, L, M, N)]

    Combine 14 rings into a product ring

28. class Tuple15Group [A, B, C, D, E, F, G, H, I, J, K, L, M, N, O] extends Group[(A, B, C, D, E, F, G, H, I, J, K, L, M, N, O)]

    Combine 15 groups into a product group

29. class Tuple15Monoid [A, B, C, D, E, F, G, H, I, J, K, L, M, N, O] extends Monoid[(A, B, C, D, E, F, G, H, I, J, K, L, M, N, O)]

    Combine 15 monoids into a product monoid

30. class Tuple15Ring [A, B, C, D, E, F, G, H, I, J, K, L, M, N, O] extends Ring[(A, B, C, D, E, F, G, H, I, J, K, L, M, N, O)]

    Combine 15 rings into a product ring

31. class Tuple16Group [A, B, C, D, E, F, G, H, I, J, K, L, M, N, O, P] extends Group[(A, B, C, D, E, F, G, H, I, J, K, L, M, N, O, P)]

    Combine 16 groups into a product group

32. class Tuple16Monoid [A, B, C, D, E, F, G, H, I, J, K, L, M, N, O, P] extends Monoid[(A, B, C, D, E, F, G, H, I, J, K, L, M, N, O, P)]

    Combine 16 monoids into a product monoid

33. class Tuple16Ring [A, B, C, D, E, F, G, H, I, J, K, L, M, N, O, P] extends Ring[(A, B, C, D, E, F, G, H, I, J, K, L, M, N, O, P)]

    Combine 16 rings into a product ring

34. class Tuple17Group [A, B, C, D, E, F, G, H, I, J, K, L, M, N, O, P, Q] extends Group[(A, B, C, D, E, F, G, H, I, J, K, L, M, N, O, P, Q)]

    Combine 17 groups into a product group

35. class Tuple17Monoid [A, B, C, D, E, F, G, H, I, J, K, L, M, N, O, P, Q] extends Monoid[(A, B, C, D, E, F, G, H, I, J, K, L, M, N, O, P, Q)]

    Combine 17 monoids into a product monoid

36. class Tuple17Ring [A, B, C, D, E, F, G, H, I, J, K, L, M, N, O, P, Q] extends Ring[(A, B, C, D, E, F, G, H, I, J, K, L, M, N, O, P, Q)]

    Combine 17 rings into a product ring

37. class Tuple18Group [A, B, C, D, E, F, G, H, I, J, K, L, M, N, O, P, Q, R] extends Group[(A, B, C, D, E, F, G, H, I, J, K, L, M, N, O, P, Q, R)]

    Combine 18 groups into a product group

38. class Tuple18Monoid [A, B, C, D, E, F, G, H, I, J, K, L, M, N, O, P, Q, R] extends Monoid[(A, B, C, D, E, F, G, H, I, J, K, L, M, N, O, P, Q, R)]

    Combine 18 monoids into a product monoid

39. class Tuple18Ring [A, B, C, D, E, F, G, H, I, J, K, L, M, N, O, P, Q, R] extends Ring[(A, B, C, D, E, F, G, H, I, J, K, L, M, N, O, P, Q, R)]

    Combine 18 rings into a product ring

40. class Tuple19Group [A, B, C, D, E, F, G, H, I, J, K, L, M, N, O, P, Q, R, S] extends Group[(A, B, C, D, E, F, G, H, I, J, K, L, M, N, O, P, Q, R, S)]

    Combine 19 groups into a product group

41. class Tuple19Monoid [A, B, C, D, E, F, G, H, I, J, K, L, M, N, O, P, Q, R, S] extends Monoid[(A, B, C, D, E, F, G, H, I, J, K, L, M, N, O, P, Q, R, S)]

    Combine 19 monoids into a product monoid

42. class Tuple19Ring [A, B, C, D, E, F, G, H, I, J, K, L, M, N, O, P, Q, R, S] extends Ring[(A, B, C, D, E, F, G, H, I, J, K, L, M, N, O, P, Q, R, S)]

    Combine 19 rings into a product ring

43. class Tuple20Group [A, B, C, D, E, F, G, H, I, J, K, L, M, N, O, P, Q, R, S, T] extends Group[(A, B, C, D, E, F, G, H, I, J, K, L, M, N, O, P, Q, R, S, T)]

    Combine 20 groups into a product group

44. class Tuple20Monoid [A, B, C, D, E, F, G, H, I, J, K, L, M, N, O, P, Q, R, S, T] extends Monoid[(A, B, C, D, E, F, G, H, I, J, K, L, M, N, O, P, Q, R, S, T)]

    Combine 20 monoids into a product monoid

45. class Tuple20Ring [A, B, C, D, E, F, G, H, I, J, K, L, M, N, O, P, Q, R, S, T] extends Ring[(A, B, C, D, E, F, G, H, I, J, K, L, M, N, O, P, Q, R, S, T)]

    Combine 20 rings into a product ring

46. class Tuple21Group [A, B, C, D, E, F, G, H, I, J, K, L, M, N, O, P, Q, R, S, T, U] extends Group[(A, B, C, D, E, F, G, H, I, J, K, L, M, N, O, P, Q, R, S, T, U)]

    Combine 21 groups into a product group

47. class Tuple21Monoid [A, B, C, D, E, F, G, H, I, J, K, L, M, N, O, P, Q, R, S, T, U] extends Monoid[(A, B, C, D, E, F, G, H, I, J, K, L, M, N, O, P, Q, R, S, T, U)]

    Combine 21 monoids into a product monoid

48. class Tuple21Ring [A, B, C, D, E, F, G, H, I, J, K, L, M, N, O, P, Q, R, S, T, U] extends Ring[(A, B, C, D, E, F, G, H, I, J, K, L, M, N, O, P, Q, R, S, T, U)]

    Combine 21 rings into a product ring

49. class Tuple22Group [A, B, C, D, E, F, G, H, I, J, K, L, M, N, O, P, Q, R, S, T, U, V] extends Group[(A, B, C, D, E, F, G, H, I, J, K, L, M, N, O, P, Q, R, S, T, U, V)]

    Combine 22 groups into a product group

50. class Tuple22Monoid [A, B, C, D, E, F, G, H, I, J, K, L, M, N, O, P, Q, R, S, T, U, V] extends Monoid[(A, B, C, D, E, F, G, H, I, J, K, L, M, N, O, P, Q, R, S, T, U, V)]

    Combine 22 monoids into a product monoid

51. class Tuple22Ring [A, B, C, D, E, F, G, H, I, J, K, L, M, N, O, P, Q, R, S, T, U, V] extends Ring[(A, B, C, D, E, F, G, H, I, J, K, L, M, N, O, P, Q, R, S, T, U, V)]

    Combine 22 rings into a product ring

52. class Tuple2Group [T, U] extends Group[(T, U)]

    Combine two groups into a product group

53. class Tuple2Monoid [T, U] extends Monoid[(T, U)]

    Combine two monoids into a product monoid

54. class Tuple2Ring [T, U] extends Ring[(T, U)]

    Combine two rings into a product ring

55. class Tuple3Group [A, B, C] extends Group[(A, B, C)]

    Combine 3 groups into a product group

56. class Tuple3Monoid [A, B, C] extends Monoid[(A, B, C)]

    Combine 3 monoids into a product monoid

57. class Tuple3Ring [A, B, C] extends Ring[(A, B, C)]

    Combine 3 rings into a product ring

58. class Tuple4Group [A, B, C, D] extends Group[(A, B, C, D)]

    Combine 4 groups into a product group

59. class Tuple4Monoid [A, B, C, D] extends Monoid[(A, B, C, D)]

    Combine 4 monoids into a product monoid

60. class Tuple4Ring [A, B, C, D] extends Ring[(A, B, C, D)]

    Combine 4 rings into a product ring

61. class Tuple5Group [A, B, C, D, E] extends Group[(A, B, C, D, E)]

    Combine 5 groups into a product group

62. class Tuple5Monoid [A, B, C, D, E] extends Monoid[(A, B, C, D, E)]

    Combine 5 monoids into a product monoid

63. class Tuple5Ring [A, B, C, D, E] extends Ring[(A, B, C, D, E)]

    Combine 5 rings into a product ring

64. class Tuple6Group [A, B, C, D, E, F] extends Group[(A, B, C, D, E, F)]

    Combine 6 groups into a product group

65. class Tuple6Monoid [A, B, C, D, E, F] extends Monoid[(A, B, C, D, E, F)]

    Combine 6 monoids into a product monoid

66. class Tuple6Ring [A, B, C, D, E, F] extends Ring[(A, B, C, D, E, F)]

    Combine 6 rings into a product ring

67. class Tuple7Group [A, B, C, D, E, F, G] extends Group[(A, B, C, D, E, F, G)]

    Combine 7 groups into a product group

68. class Tuple7Monoid [A, B, C, D, E, F, G] extends Monoid[(A, B, C, D, E, F, G)]

    Combine 7 monoids into a product monoid

69. class Tuple7Ring [A, B, C, D, E, F, G] extends Ring[(A, B, C, D, E, F, G)]

    Combine 7 rings into a product ring

70. class Tuple8Group [A, B, C, D, E, F, G, H] extends Group[(A, B, C, D, E, F, G, H)]

    Combine 8 groups into a product group

71. class Tuple8Monoid [A, B, C, D, E, F, G, H] extends Monoid[(A, B, C, D, E, F, G, H)]

    Combine 8 monoids into a product monoid

72. class Tuple8Ring [A, B, C, D, E, F, G, H] extends Ring[(A, B, C, D, E, F, G, H)]

    Combine 8 rings into a product ring

73. class Tuple9Group [A, B, C, D, E, F, G, H, I] extends Group[(A, B, C, D, E, F, G, H, I)]

    Combine 9 groups into a product group

74. class Tuple9Monoid [A, B, C, D, E, F, G, H, I] extends Monoid[(A, B, C, D, E, F, G, H, I)]

    Combine 9 monoids into a product monoid

75. class Tuple9Ring [A, B, C, D, E, F, G, H, I] extends Ring[(A, B, C, D, E, F, G, H, I)]

    Combine 9 rings into a product ring