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Catenary degrees of elements in numerical monoids Christopher ONeill Texas A&M University coneill@math.tamu.edu Joint with Vadim Ponomarenko, Reuben Tate*, and Gautam Webb* January 11, 2015 Christopher ONeill (Texas A&M

  1. Numerical monoids Definition A numerical monoid S is an additive submonoid of N with | N \ S | < ∞ . Example Let S = � 2 , 3 � = { 2 , 3 , 4 , 5 , . . . } under addition . C [ S ] = C [ x 2 , x 3 ]. x 6 = x 3 · x 3 = x 2 · x 2 · x 2 6 = 3 + 3 = 2 + 2 + 2. � Factorizations are additive! Christopher O’Neill (Texas A&M University) Catenary degrees in numerical monoids January 11, 2015 3 / 14

  2. Numerical monoids Definition A numerical monoid S is an additive submonoid of N with | N \ S | < ∞ . Example Let S = � 2 , 3 � = { 2 , 3 , 4 , 5 , . . . } under addition . C [ S ] = C [ x 2 , x 3 ]. x 6 = x 3 · x 3 = x 2 · x 2 · x 2 6 = 3 + 3 = 2 + 2 + 2. � Factorizations are additive! Example McN = � 6 , 9 , 20 � = { 0 , 6 , 9 , 12 , 15 , 18 , 20 , 21 , . . . } . Christopher O’Neill (Texas A&M University) Catenary degrees in numerical monoids January 11, 2015 3 / 14

  3. Numerical monoids Definition A numerical monoid S is an additive submonoid of N with | N \ S | < ∞ . Example Let S = � 2 , 3 � = { 2 , 3 , 4 , 5 , . . . } under addition . C [ S ] = C [ x 2 , x 3 ]. x 6 = x 3 · x 3 = x 2 · x 2 · x 2 6 = 3 + 3 = 2 + 2 + 2. � Factorizations are additive! Example McN = � 6 , 9 , 20 � = { 0 , 6 , 9 , 12 , 15 , 18 , 20 , 21 , . . . } . “McNugget Monoid” Christopher O’Neill (Texas A&M University) Catenary degrees in numerical monoids January 11, 2015 3 / 14

  4. Numerical monoids Definition A numerical monoid S is an additive submonoid of N with | N \ S | < ∞ . Example Let S = � 2 , 3 � = { 2 , 3 , 4 , 5 , . . . } under addition . C [ S ] = C [ x 2 , x 3 ]. x 6 = x 3 · x 3 = x 2 · x 2 · x 2 6 = 3 + 3 = 2 + 2 + 2. � Factorizations are additive! Example McN = � 6 , 9 , 20 � = { 0 , 6 , 9 , 12 , 15 , 18 , 20 , 21 , . . . } . “McNugget Monoid” 60 = 7(6) + 2(9) Christopher O’Neill (Texas A&M University) Catenary degrees in numerical monoids January 11, 2015 3 / 14

  5. Numerical monoids Definition A numerical monoid S is an additive submonoid of N with | N \ S | < ∞ . Example Let S = � 2 , 3 � = { 2 , 3 , 4 , 5 , . . . } under addition . C [ S ] = C [ x 2 , x 3 ]. x 6 = x 3 · x 3 = x 2 · x 2 · x 2 6 = 3 + 3 = 2 + 2 + 2. � Factorizations are additive! Example McN = � 6 , 9 , 20 � = { 0 , 6 , 9 , 12 , 15 , 18 , 20 , 21 , . . . } . “McNugget Monoid” 60 = 7(6) + 2(9) = 3(20) Christopher O’Neill (Texas A&M University) Catenary degrees in numerical monoids January 11, 2015 3 / 14

  6. Numerical monoids Definition A numerical monoid S is an additive submonoid of N with | N \ S | < ∞ . Example Let S = � 2 , 3 � = { 2 , 3 , 4 , 5 , . . . } under addition . C [ S ] = C [ x 2 , x 3 ]. x 6 = x 3 · x 3 = x 2 · x 2 · x 2 6 = 3 + 3 = 2 + 2 + 2. � Factorizations are additive! Example McN = � 6 , 9 , 20 � = { 0 , 6 , 9 , 12 , 15 , 18 , 20 , 21 , . . . } . “McNugget Monoid” 60 = 7(6) + 2(9) (7 , 2 , 0) � = 3(20) (0 , 0 , 3) � Christopher O’Neill (Texas A&M University) Catenary degrees in numerical monoids January 11, 2015 3 / 14

  7. Factorization invariants: towards the catenary degree Christopher O’Neill (Texas A&M University) Catenary degrees in numerical monoids January 11, 2015 4 / 14

  8. Factorization invariants: towards the catenary degree Definition Fix a numerical monoid S = � n 1 , . . . , n k � . For n ∈ S , ( a 1 , . . . , a k ) ∈ N k : n = a 1 n 1 + · · · + a k n k � � Z S ( n ) = denotes the set of factorizations of n . Christopher O’Neill (Texas A&M University) Catenary degrees in numerical monoids January 11, 2015 4 / 14

  9. Factorization invariants: towards the catenary degree Definition Fix a numerical monoid S = � n 1 , . . . , n k � . For n ∈ S , ( a 1 , . . . , a k ) ∈ N k : n = a 1 n 1 + · · · + a k n k � � Z S ( n ) = denotes the set of factorizations of n . Equivalently, if φ : N k − → S � e i �− → n i then Z S ( n ) = φ − 1 ( n ). Christopher O’Neill (Texas A&M University) Catenary degrees in numerical monoids January 11, 2015 4 / 14

  10. Factorization invariants: towards the catenary degree Definition Fix a numerical monoid S = � n 1 , . . . , n k � . For n ∈ S , ( a 1 , . . . , a k ) ∈ N k : n = a 1 n 1 + · · · + a k n k � � Z S ( n ) = denotes the set of factorizations of n . Equivalently, if φ : N k − → S � e i �− → n i then Z S ( n ) = φ − 1 ( n ). For f , f ′ ∈ Z S ( n ), | f | = f 1 + · · · + f k Christopher O’Neill (Texas A&M University) Catenary degrees in numerical monoids January 11, 2015 4 / 14

  11. Factorization invariants: towards the catenary degree Definition Fix a numerical monoid S = � n 1 , . . . , n k � . For n ∈ S , ( a 1 , . . . , a k ) ∈ N k : n = a 1 n 1 + · · · + a k n k � � Z S ( n ) = denotes the set of factorizations of n . Equivalently, if φ : N k − → S e i � �− → n i then Z S ( n ) = φ − 1 ( n ). For f , f ′ ∈ Z S ( n ), | f | = f 1 + · · · + f k ( length of f ) Christopher O’Neill (Texas A&M University) Catenary degrees in numerical monoids January 11, 2015 4 / 14

  12. Factorization invariants: towards the catenary degree Definition Fix a numerical monoid S = � n 1 , . . . , n k � . For n ∈ S , ( a 1 , . . . , a k ) ∈ N k : n = a 1 n 1 + · · · + a k n k � � Z S ( n ) = denotes the set of factorizations of n . Equivalently, if φ : N k − → S � e i �− → n i then Z S ( n ) = φ − 1 ( n ). For f , f ′ ∈ Z S ( n ), | f | = f 1 + · · · + f k ( length of f ) gcd( f , f ′ ) = (min( f 1 , f ′ 1 ) , . . . , min( f k , f ′ k )) Christopher O’Neill (Texas A&M University) Catenary degrees in numerical monoids January 11, 2015 4 / 14

  13. Factorization invariants: towards the catenary degree Definition Fix a numerical monoid S = � n 1 , . . . , n k � . For n ∈ S , ( a 1 , . . . , a k ) ∈ N k : n = a 1 n 1 + · · · + a k n k � � Z S ( n ) = denotes the set of factorizations of n . Equivalently, if φ : N k − → S � e i �− → n i then Z S ( n ) = φ − 1 ( n ). For f , f ′ ∈ Z S ( n ), | f | = f 1 + · · · + f k ( length of f ) gcd( f , f ′ ) = (min( f 1 , f ′ 1 ) , . . . , min( f k , f ′ k )) max {| f − gcd( f , f ′ ) | , | f ′ − gcd( f , f ′ ) |} d ( f , f ′ ) = Christopher O’Neill (Texas A&M University) Catenary degrees in numerical monoids January 11, 2015 4 / 14

  14. Factorization invariants: towards the catenary degree Example S = � 4 , 6 , 7 � ⊂ N , Christopher O’Neill (Texas A&M University) Catenary degrees in numerical monoids January 11, 2015 5 / 14

  15. Factorization invariants: towards the catenary degree Example S = � 4 , 6 , 7 � ⊂ N , f = (3 , 1 , 1) , f ′ = (1 , 0 , 3) ∈ Z S (25). Christopher O’Neill (Texas A&M University) Catenary degrees in numerical monoids January 11, 2015 5 / 14

  16. Factorization invariants: towards the catenary degree Example S = � 4 , 6 , 7 � ⊂ N , f = (3 , 1 , 1) , f ′ = (1 , 0 , 3) ∈ Z S (25). 4 7 6 4 7 7 4 4 7 (3,1,1) (1,0,3) Christopher O’Neill (Texas A&M University) Catenary degrees in numerical monoids January 11, 2015 5 / 14

  17. Factorization invariants: towards the catenary degree Example S = � 4 , 6 , 7 � ⊂ N , f = (3 , 1 , 1) , f ′ = (1 , 0 , 3) ∈ Z S (25). g = gcd( f , f ′ ) 4 7 6 4 7 7 4 4 7 (3,1,1) (1,0,3) Christopher O’Neill (Texas A&M University) Catenary degrees in numerical monoids January 11, 2015 5 / 14

  18. Factorization invariants: towards the catenary degree Example S = � 4 , 6 , 7 � ⊂ N , f = (3 , 1 , 1) , f ′ = (1 , 0 , 3) ∈ Z S (25). g = gcd( f , f ′ ) 4 7 6 4 7 7 4 4 7 (3,1,1) (1,0,3) Christopher O’Neill (Texas A&M University) Catenary degrees in numerical monoids January 11, 2015 5 / 14

  19. Factorization invariants: towards the catenary degree Example S = � 4 , 6 , 7 � ⊂ N , f = (3 , 1 , 1) , f ′ = (1 , 0 , 3) ∈ Z S (25). g = gcd( f , f ′ ) = (1 , 0 , 1) . 4 7 6 4 7 7 4 4 7 (3,1,1) (1,0,3) Christopher O’Neill (Texas A&M University) Catenary degrees in numerical monoids January 11, 2015 5 / 14

  20. Factorization invariants: towards the catenary degree Example S = � 4 , 6 , 7 � ⊂ N , f = (3 , 1 , 1) , f ′ = (1 , 0 , 3) ∈ Z S (25). g = gcd( f , f ′ ) = (1 , 0 , 1) . d ( f , f ′ ) 4 7 6 4 7 7 4 4 7 (3,1,1) (1,0,3) Christopher O’Neill (Texas A&M University) Catenary degrees in numerical monoids January 11, 2015 5 / 14

  21. Factorization invariants: towards the catenary degree Example S = � 4 , 6 , 7 � ⊂ N , f = (3 , 1 , 1) , f ′ = (1 , 0 , 3) ∈ Z S (25). g = gcd( f , f ′ ) = (1 , 0 , 1) . d ( f , f ′ ) = max {| f − g | , | f ′ − g |} 4 7 6 4 7 7 4 4 7 (3,1,1) (1,0,3) Christopher O’Neill (Texas A&M University) Catenary degrees in numerical monoids January 11, 2015 5 / 14

  22. Factorization invariants: towards the catenary degree Example S = � 4 , 6 , 7 � ⊂ N , f = (3 , 1 , 1) , f ′ = (1 , 0 , 3) ∈ Z S (25). g = gcd( f , f ′ ) = (1 , 0 , 1) . d ( f , f ′ ) = max {| f − g | , | f ′ − g |} 4 7 6 4 7 7 4 4 7 (3,1,1) (1,0,3) Christopher O’Neill (Texas A&M University) Catenary degrees in numerical monoids January 11, 2015 5 / 14

  23. Factorization invariants: towards the catenary degree Example S = � 4 , 6 , 7 � ⊂ N , f = (3 , 1 , 1) , f ′ = (1 , 0 , 3) ∈ Z S (25). g = gcd( f , f ′ ) = (1 , 0 , 1) . d ( f , f ′ ) = max {| f − g | , | f ′ − g |} = 3 . 4 7 6 4 7 7 4 4 7 (3,1,1) (1,0,3) Christopher O’Neill (Texas A&M University) Catenary degrees in numerical monoids January 11, 2015 5 / 14

  24. The catenary degree Definition Fix a numerical monoid S = � n 1 , . . . , n k � . For n ∈ S , define the catenary degree c ( n ) as follows: Christopher O’Neill (Texas A&M University) Catenary degrees in numerical monoids January 11, 2015 6 / 14

  25. The catenary degree Definition Fix a numerical monoid S = � n 1 , . . . , n k � . For n ∈ S , define the catenary degree c ( n ) as follows: 1 Construct a complete graph G with vertex set Z S ( n ) where each edge ( f , f ′ ) has label d ( f , f ′ ) ( catenary graph ). Christopher O’Neill (Texas A&M University) Catenary degrees in numerical monoids January 11, 2015 6 / 14

  26. The catenary degree Definition Fix a numerical monoid S = � n 1 , . . . , n k � . For n ∈ S , define the catenary degree c ( n ) as follows: 1 Construct a complete graph G with vertex set Z S ( n ) where each edge ( f , f ′ ) has label d ( f , f ′ ) ( catenary graph ). 2 Locate the largest edge weight e in G . Christopher O’Neill (Texas A&M University) Catenary degrees in numerical monoids January 11, 2015 6 / 14

  27. The catenary degree Definition Fix a numerical monoid S = � n 1 , . . . , n k � . For n ∈ S , define the catenary degree c ( n ) as follows: 1 Construct a complete graph G with vertex set Z S ( n ) where each edge ( f , f ′ ) has label d ( f , f ′ ) ( catenary graph ). 2 Locate the largest edge weight e in G . 3 Remove all edges from G with weight e . Christopher O’Neill (Texas A&M University) Catenary degrees in numerical monoids January 11, 2015 6 / 14

  28. The catenary degree Definition Fix a numerical monoid S = � n 1 , . . . , n k � . For n ∈ S , define the catenary degree c ( n ) as follows: 1 Construct a complete graph G with vertex set Z S ( n ) where each edge ( f , f ′ ) has label d ( f , f ′ ) ( catenary graph ). 2 Locate the largest edge weight e in G . 3 Remove all edges from G with weight e . 4 If G is disconnected, return e . Otherwise, return to step 2. Christopher O’Neill (Texas A&M University) Catenary degrees in numerical monoids January 11, 2015 6 / 14

  29. The catenary degree Definition Fix a numerical monoid S = � n 1 , . . . , n k � . For n ∈ S , define the catenary degree c ( n ) as follows: 1 Construct a complete graph G with vertex set Z S ( n ) where each edge ( f , f ′ ) has label d ( f , f ′ ) ( catenary graph ). 2 Locate the largest edge weight e in G . 3 Remove all edges from G with weight e . 4 If G is disconnected, return e . Otherwise, return to step 2. If | Z S ( n ) | = 1, define c ( n ) = 0. Christopher O’Neill (Texas A&M University) Catenary degrees in numerical monoids January 11, 2015 6 / 14

  30. A Big Example S = � 11 , 36 , 39 � , n = 450 Christopher O’Neill (Texas A&M University) Catenary degrees in numerical monoids January 11, 2015 7 / 14

  31. A Big Example S = � 11 , 36 , 39 � , n = 450 (0,6,6) (27,1,3) 27 4 4 24 24 (3,4,7) (24,3,2) 8 8 21 21 21 12 19 12 4 4 18 18 18 8 (6,2,8) (21,5,1) 8 17 17 4 4 16 (9,0,9) (18,7,0) Christopher O’Neill (Texas A&M University) Catenary degrees in numerical monoids January 11, 2015 7 / 14

  32. A Big Example S = � 11 , 36 , 39 � , n = 450 (0,6,6) (27,1,3) 27 4 4 24 24 (3,4,7) (24,3,2) 8 8 21 21 21 12 19 12 4 4 18 18 18 8 (6,2,8) (21,5,1) 8 17 17 4 4 16 (9,0,9) (18,7,0) Christopher O’Neill (Texas A&M University) Catenary degrees in numerical monoids January 11, 2015 7 / 14

  33. A Big Example S = � 11 , 36 , 39 � , n = 450 (0,6,6) (27,1,3) 4 4 24 24 (3,4,7) (24,3,2) 8 8 21 21 21 12 19 12 4 4 18 18 18 8 (6,2,8) (21,5,1) 8 17 17 4 4 16 (9,0,9) (18,7,0) Christopher O’Neill (Texas A&M University) Catenary degrees in numerical monoids January 11, 2015 7 / 14

  34. A Big Example S = � 11 , 36 , 39 � , n = 450 (0,6,6) (27,1,3) 4 4 24 24 (3,4,7) (24,3,2) 8 8 21 21 21 12 19 12 4 4 18 18 18 8 (6,2,8) (21,5,1) 8 17 17 4 4 16 (9,0,9) (18,7,0) Christopher O’Neill (Texas A&M University) Catenary degrees in numerical monoids January 11, 2015 7 / 14

  35. A Big Example S = � 11 , 36 , 39 � , n = 450 (0,6,6) (27,1,3) 4 4 (3,4,7) (24,3,2) 8 8 21 21 21 12 19 12 4 4 18 18 18 8 (6,2,8) (21,5,1) 8 17 17 4 4 16 (9,0,9) (18,7,0) Christopher O’Neill (Texas A&M University) Catenary degrees in numerical monoids January 11, 2015 7 / 14

  36. A Big Example S = � 11 , 36 , 39 � , n = 450 (0,6,6) (27,1,3) 4 4 (3,4,7) (24,3,2) 8 8 21 21 21 12 19 12 4 4 18 18 18 8 (6,2,8) (21,5,1) 8 17 17 4 4 16 (9,0,9) (18,7,0) Christopher O’Neill (Texas A&M University) Catenary degrees in numerical monoids January 11, 2015 7 / 14

  37. A Big Example S = � 11 , 36 , 39 � , n = 450 (0,6,6) (27,1,3) 4 4 (3,4,7) (24,3,2) 8 8 12 19 12 4 4 18 18 18 8 (6,2,8) (21,5,1) 8 17 17 4 4 16 (9,0,9) (18,7,0) Christopher O’Neill (Texas A&M University) Catenary degrees in numerical monoids January 11, 2015 7 / 14

  38. A Big Example S = � 11 , 36 , 39 � , n = 450 (0,6,6) (27,1,3) 4 4 (3,4,7) (24,3,2) 8 8 12 19 12 4 4 18 18 18 8 (6,2,8) (21,5,1) 8 17 17 4 4 16 (9,0,9) (18,7,0) Christopher O’Neill (Texas A&M University) Catenary degrees in numerical monoids January 11, 2015 7 / 14

  39. A Big Example S = � 11 , 36 , 39 � , n = 450 (0,6,6) (27,1,3) 4 4 (3,4,7) (24,3,2) 8 8 12 12 4 4 18 18 18 8 (6,2,8) (21,5,1) 8 17 17 4 4 16 (9,0,9) (18,7,0) Christopher O’Neill (Texas A&M University) Catenary degrees in numerical monoids January 11, 2015 7 / 14

  40. A Big Example S = � 11 , 36 , 39 � , n = 450 (0,6,6) (27,1,3) 4 4 (3,4,7) (24,3,2) 8 8 12 12 4 4 18 18 18 8 (6,2,8) (21,5,1) 8 17 17 4 4 16 (9,0,9) (18,7,0) Christopher O’Neill (Texas A&M University) Catenary degrees in numerical monoids January 11, 2015 7 / 14

  41. A Big Example S = � 11 , 36 , 39 � , n = 450 (0,6,6) (27,1,3) 4 4 (3,4,7) (24,3,2) 8 8 12 12 4 4 8 (6,2,8) (21,5,1) 8 17 17 4 4 16 (9,0,9) (18,7,0) Christopher O’Neill (Texas A&M University) Catenary degrees in numerical monoids January 11, 2015 7 / 14

  42. A Big Example S = � 11 , 36 , 39 � , n = 450 (0,6,6) (27,1,3) 4 4 (3,4,7) (24,3,2) 8 8 12 12 4 4 8 (6,2,8) (21,5,1) 8 17 17 4 4 16 (9,0,9) (18,7,0) Christopher O’Neill (Texas A&M University) Catenary degrees in numerical monoids January 11, 2015 7 / 14

  43. A Big Example S = � 11 , 36 , 39 � , n = 450 (0,6,6) (27,1,3) 4 4 (3,4,7) (24,3,2) 8 8 12 12 4 4 8 (6,2,8) (21,5,1) 8 4 4 16 (9,0,9) (18,7,0) Christopher O’Neill (Texas A&M University) Catenary degrees in numerical monoids January 11, 2015 7 / 14

  44. A Big Example S = � 11 , 36 , 39 � , n = 450 (0,6,6) (27,1,3) 4 4 (3,4,7) (24,3,2) 8 8 12 12 4 4 8 (6,2,8) (21,5,1) 8 4 4 16 (9,0,9) (18,7,0) Christopher O’Neill (Texas A&M University) Catenary degrees in numerical monoids January 11, 2015 7 / 14

  45. A Big Example S = � 11 , 36 , 39 � , n = 450 (0,6,6) (27,1,3) 4 4 (3,4,7) (24,3,2) 8 8 12 12 4 4 8 (6,2,8) (21,5,1) 8 4 4 (9,0,9) (18,7,0) Christopher O’Neill (Texas A&M University) Catenary degrees in numerical monoids January 11, 2015 7 / 14

  46. A Big Example S = � 11 , 36 , 39 � , n = 450, c ( n ) = 16 (0,6,6) (27,1,3) 4 4 (3,4,7) (24,3,2) 8 8 12 12 4 4 8 (6,2,8) (21,5,1) 8 4 4 (9,0,9) (18,7,0) Christopher O’Neill (Texas A&M University) Catenary degrees in numerical monoids January 11, 2015 7 / 14

  47. A Big Example, Method 2 S = � 11 , 36 , 39 � , n = 450 Christopher O’Neill (Texas A&M University) Catenary degrees in numerical monoids January 11, 2015 8 / 14

  48. A Big Example, Method 2 S = � 11 , 36 , 39 � , n = 450 (0,6,6) (27,1,3) (3,4,7) (24,3,2) (6,2,8) (21,5,1) (9,0,9) (18,7,0) Christopher O’Neill (Texas A&M University) Catenary degrees in numerical monoids January 11, 2015 8 / 14

  49. A Big Example, Method 2 S = � 11 , 36 , 39 � , n = 450 (0,6,6) (27,1,3) 4 4 (3,4,7) (24,3,2) 4 4 (6,2,8) (21,5,1) 4 4 (9,0,9) (18,7,0) Christopher O’Neill (Texas A&M University) Catenary degrees in numerical monoids January 11, 2015 8 / 14

  50. A Big Example, Method 2 S = � 11 , 36 , 39 � , n = 450 (0,6,6) (27,1,3) 4 4 (3,4,7) (24,3,2) 8 8 4 4 8 (6,2,8) (21,5,1) 8 4 4 (9,0,9) (18,7,0) Christopher O’Neill (Texas A&M University) Catenary degrees in numerical monoids January 11, 2015 8 / 14

  51. A Big Example, Method 2 S = � 11 , 36 , 39 � , n = 450 (0,6,6) (27,1,3) 4 4 (3,4,7) (24,3,2) 8 8 12 12 4 4 8 (6,2,8) (21,5,1) 8 4 4 (9,0,9) (18,7,0) Christopher O’Neill (Texas A&M University) Catenary degrees in numerical monoids January 11, 2015 8 / 14

  52. A Big Example, Method 2 S = � 11 , 36 , 39 � , n = 450 (0,6,6) (27,1,3) 4 4 (3,4,7) (24,3,2) 8 8 12 12 4 4 8 (6,2,8) (21,5,1) 8 4 4 16 (9,0,9) (18,7,0) Christopher O’Neill (Texas A&M University) Catenary degrees in numerical monoids January 11, 2015 8 / 14

  53. A Big Example, Method 2 S = � 11 , 36 , 39 � , n = 450, c ( n ) = 16 (0,6,6) (27,1,3) 4 4 (3,4,7) (24,3,2) 8 8 12 12 4 4 8 (6,2,8) (21,5,1) 8 4 4 16 (9,0,9) (18,7,0) Christopher O’Neill (Texas A&M University) Catenary degrees in numerical monoids January 11, 2015 8 / 14

  54. Betti elements Definition For an element n ∈ S = � n 1 , . . . , n k � , let ∇ n denote the subgraph of the catenary graph in which only edges ( f , f ′ ) with gcd( f , f ′ ) � = 0 are drawn. Christopher O’Neill (Texas A&M University) Catenary degrees in numerical monoids January 11, 2015 9 / 14

  55. Betti elements Definition For an element n ∈ S = � n 1 , . . . , n k � , let ∇ n denote the subgraph of the catenary graph in which only edges ( f , f ′ ) with gcd( f , f ′ ) � = 0 are drawn. We say n is a Betti element of S if ∇ n is disconnected. Christopher O’Neill (Texas A&M University) Catenary degrees in numerical monoids January 11, 2015 9 / 14

  56. Betti elements Definition For an element n ∈ S = � n 1 , . . . , n k � , let ∇ n denote the subgraph of the catenary graph in which only edges ( f , f ′ ) with gcd( f , f ′ ) � = 0 are drawn. We say n is a Betti element of S if ∇ n is disconnected. Example S = � 10 , 15 , 17 � has Betti elements 30 and 85. Christopher O’Neill (Texas A&M University) Catenary degrees in numerical monoids January 11, 2015 9 / 14

  57. Betti elements Definition For an element n ∈ S = � n 1 , . . . , n k � , let ∇ n denote the subgraph of the catenary graph in which only edges ( f , f ′ ) with gcd( f , f ′ ) � = 0 are drawn. We say n is a Betti element of S if ∇ n is disconnected. Example S = � 10 , 15 , 17 � has Betti elements 30 and 85. ∇ 30 : ∇ 85 : (0,0,5) (3,0,0) (0,2,0) (7,1,0) (1,5,0) (4,3,0) Christopher O’Neill (Texas A&M University) Catenary degrees in numerical monoids January 11, 2015 9 / 14

  58. Maximal catenary degree in S Theorem max { c ( n ) : n ∈ S } = max { c ( b ) : b Betti element of S } . Christopher O’Neill (Texas A&M University) Catenary degrees in numerical monoids January 11, 2015 10 / 14

  59. Maximal catenary degree in S Theorem max { c ( n ) : n ∈ S } = max { c ( b ) : b Betti element of S } . Key concept: Cover morphisms. Christopher O’Neill (Texas A&M University) Catenary degrees in numerical monoids January 11, 2015 10 / 14

  60. Maximal catenary degree in S Theorem max { c ( n ) : n ∈ S } = max { c ( b ) : b Betti element of S } . Key concept: Cover morphisms. Z S ( n ) Z S ( n + n i ) Christopher O’Neill (Texas A&M University) Catenary degrees in numerical monoids January 11, 2015 10 / 14

  61. Maximal catenary degree in S Theorem max { c ( n ) : n ∈ S } = max { c ( b ) : b Betti element of S } . Key concept: Cover morphisms. Z S ( n ) ֒ − − − − − − − − − − − → Z S ( n + n i ) Christopher O’Neill (Texas A&M University) Catenary degrees in numerical monoids January 11, 2015 10 / 14

  62. Maximal catenary degree in S Theorem max { c ( n ) : n ∈ S } = max { c ( b ) : b Betti element of S } . Key concept: Cover morphisms. Z S ( n ) ֒ − − − − − − − − − − − → Z S ( n + n i ) f Christopher O’Neill (Texas A&M University) Catenary degrees in numerical monoids January 11, 2015 10 / 14

  63. Maximal catenary degree in S Theorem max { c ( n ) : n ∈ S } = max { c ( b ) : b Betti element of S } . Key concept: Cover morphisms. Z S ( n ) ֒ − − − − − − − − − − − → Z S ( n + n i ) f �− − − − − − − − − − − − → f + e i Christopher O’Neill (Texas A&M University) Catenary degrees in numerical monoids January 11, 2015 10 / 14

  64. Maximal catenary degree in S Theorem max { c ( n ) : n ∈ S } = max { c ( b ) : b Betti element of S } . Key concept: Cover morphisms. 3 2 3 5 Z S ( n ) ֒ − − − − − − − − − − − → Z S ( n + n i ) f �− − − − − − − − − − − − → f + e i Christopher O’Neill (Texas A&M University) Catenary degrees in numerical monoids January 11, 2015 10 / 14

  65. Maximal catenary degree in S Theorem max { c ( n ) : n ∈ S } = max { c ( b ) : b Betti element of S } . Key concept: Cover morphisms. 3 3 2 2 3 5 3 5 Z S ( n ) ֒ − − − − − − − − − − − → Z S ( n + n i ) f �− − − − − − − − − − − − → f + e i Christopher O’Neill (Texas A&M University) Catenary degrees in numerical monoids January 11, 2015 10 / 14

  66. Maximal catenary degree in S Theorem max { c ( n ) : n ∈ S } = max { c ( b ) : b Betti element of S } . Christopher O’Neill (Texas A&M University) Catenary degrees in numerical monoids January 11, 2015 11 / 14

  67. Maximal catenary degree in S Theorem max { c ( n ) : n ∈ S } = max { c ( b ) : b Betti element of S } . Idea for proof: Certain edges (determined by Betti elements) connect the catenary graph of each n ∈ S . Christopher O’Neill (Texas A&M University) Catenary degrees in numerical monoids January 11, 2015 11 / 14

  68. Maximal catenary degree in S Theorem max { c ( n ) : n ∈ S } = max { c ( b ) : b Betti element of S } . Idea for proof: Certain edges (determined by Betti elements) connect the catenary graph of each n ∈ S . Christopher O’Neill (Texas A&M University) Catenary degrees in numerical monoids January 11, 2015 11 / 14

  69. Maximal catenary degree in S Theorem max { c ( n ) : n ∈ S } = max { c ( b ) : b Betti element of S } . Idea for proof: Certain edges (determined by Betti elements) connect the catenary graph of each n ∈ S . Christopher O’Neill (Texas A&M University) Catenary degrees in numerical monoids January 11, 2015 11 / 14

  70. Maximal catenary degree in S Theorem max { c ( n ) : n ∈ S } = max { c ( b ) : b Betti element of S } . Idea for proof: Certain edges (determined by Betti elements) connect the catenary graph of each n ∈ S . Christopher O’Neill (Texas A&M University) Catenary degrees in numerical monoids January 11, 2015 11 / 14

  71. Maximal catenary degree in S Theorem max { c ( n ) : n ∈ S } = max { c ( b ) : b Betti element of S } . Idea for proof: Certain edges (determined by Betti elements) connect the catenary graph of each n ∈ S . Christopher O’Neill (Texas A&M University) Catenary degrees in numerical monoids January 11, 2015 11 / 14

  72. Minimal (nonzero) catenary degree in S Conjecture min { c ( n ) > 0 : n ∈ S } = min { c ( b ) : b Betti element of S } . Christopher O’Neill (Texas A&M University) Catenary degrees in numerical monoids January 11, 2015 12 / 14

  73. Minimal (nonzero) catenary degree in S Conjecture Theorem (O., Ponomarenko, Tate, Webb) min { c ( n ) > 0 : n ∈ S } = min { c ( b ) : b Betti element of S } . Christopher O’Neill (Texas A&M University) Catenary degrees in numerical monoids January 11, 2015 12 / 14

  74. Minimal (nonzero) catenary degree in S Conjecture Theorem (O., Ponomarenko, Tate, Webb) min { c ( n ) > 0 : n ∈ S } = min { c ( b ) : b Betti element of S } . B = min { c ( b ) : b Betti element of S } . Christopher O’Neill (Texas A&M University) Catenary degrees in numerical monoids January 11, 2015 12 / 14

  75. Minimal (nonzero) catenary degree in S Conjecture Theorem (O., Ponomarenko, Tate, Webb) min { c ( n ) > 0 : n ∈ S } = min { c ( b ) : b Betti element of S } . B = min { c ( b ) : b Betti element of S } . Lemma If f , f ′ ∈ Z S ( n ) Christopher O’Neill (Texas A&M University) Catenary degrees in numerical monoids January 11, 2015 12 / 14

  76. Minimal (nonzero) catenary degree in S Conjecture Theorem (O., Ponomarenko, Tate, Webb) min { c ( n ) > 0 : n ∈ S } = min { c ( b ) : b Betti element of S } . B = min { c ( b ) : b Betti element of S } . Lemma If f , f ′ ∈ Z S ( n ) and d ( f , f ′ ) < B, Christopher O’Neill (Texas A&M University) Catenary degrees in numerical monoids January 11, 2015 12 / 14

  77. Minimal (nonzero) catenary degree in S Conjecture Theorem (O., Ponomarenko, Tate, Webb) min { c ( n ) > 0 : n ∈ S } = min { c ( b ) : b Betti element of S } . B = min { c ( b ) : b Betti element of S } . Lemma If f , f ′ ∈ Z S ( n ) and d ( f , f ′ ) < B, then there exists f ′′ ∈ Z S ( n ) Christopher O’Neill (Texas A&M University) Catenary degrees in numerical monoids January 11, 2015 12 / 14

  78. Minimal (nonzero) catenary degree in S Conjecture Theorem (O., Ponomarenko, Tate, Webb) min { c ( n ) > 0 : n ∈ S } = min { c ( b ) : b Betti element of S } . B = min { c ( b ) : b Betti element of S } . Lemma If f , f ′ ∈ Z S ( n ) and d ( f , f ′ ) < B, then there exists f ′′ ∈ Z S ( n ) with max {| f | , | f ′ |} < | f ′′ | . Christopher O’Neill (Texas A&M University) Catenary degrees in numerical monoids January 11, 2015 12 / 14

  79. Minimal (nonzero) catenary degree in S Conjecture Theorem (O., Ponomarenko, Tate, Webb) min { c ( n ) > 0 : n ∈ S } = min { c ( b ) : b Betti element of S } . Christopher O’Neill (Texas A&M University) Catenary degrees in numerical monoids January 11, 2015 12 / 14

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