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Computing the delta set and -primality in numerical monoids Christopher ONeill Texas A&M University coneill@math.tamu.edu Joint with Thomas Barron and Roberto Pelayo June 13, 2015 Christopher ONeill (Texas A&M

  1. A solution: dynamic programming Fix n ∈ S = � n 1 , . . . , n k � . For each i ≤ k , φ i : Z( n − n i ) − → Z( n ) a �− → a + e i Christopher O’Neill (Texas A&M University)Computing the delta set and ω -primality in numerical monoidsJune 13, 2015 4 / 13

  2. A solution: dynamic programming Fix n ∈ S = � n 1 , . . . , n k � . For each i ≤ k , φ i : Z( n − n i ) − → Z( n ) a �− → a + e i S = � 6 , 9 , 20 � : Christopher O’Neill (Texas A&M University)Computing the delta set and ω -primality in numerical monoidsJune 13, 2015 4 / 13

  3. A solution: dynamic programming Fix n ∈ S = � n 1 , . . . , n k � . For each i ≤ k , φ i : Z( n − n i ) − → Z( n ) a �− → a + e i S = � 6 , 9 , 20 � : Christopher O’Neill (Texas A&M University)Computing the delta set and ω -primality in numerical monoidsJune 13, 2015 4 / 13

  4. A solution: dynamic programming Fix n ∈ S = � n 1 , . . . , n k � . For each i ≤ k , φ i : Z( n − n i ) − → Z( n ) a �− → a + e i S = � 6 , 9 , 20 � : Christopher O’Neill (Texas A&M University)Computing the delta set and ω -primality in numerical monoidsJune 13, 2015 4 / 13

  5. A solution: dynamic programming Fix n ∈ S = � n 1 , . . . , n k � . For each i ≤ k , φ i : Z( n − n i ) − → Z( n ) a �− → a + e i S = � 6 , 9 , 20 � : Christopher O’Neill (Texas A&M University)Computing the delta set and ω -primality in numerical monoidsJune 13, 2015 4 / 13

  6. A solution: dynamic programming Fix n ∈ S = � n 1 , . . . , n k � . For each i ≤ k , φ i : Z( n − n i ) − → Z( n ) a �− → a + e i S = � 6 , 9 , 20 � : Christopher O’Neill (Texas A&M University)Computing the delta set and ω -primality in numerical monoidsJune 13, 2015 4 / 13

  7. A solution: dynamic programming Fix n ∈ S = � n 1 , . . . , n k � . For each i ≤ k , φ i : Z( n − n i ) − → Z( n ) a �− → a + e i Z( n ) = � i ≤ k φ i (Z( n − n i )) S = � 6 , 9 , 20 � : Christopher O’Neill (Texas A&M University)Computing the delta set and ω -primality in numerical monoidsJune 13, 2015 4 / 13

  8. A solution: dynamic programming Fix n ∈ S = � n 1 , . . . , n k � . For each i ≤ k , φ i : Z( n − n i ) − → Z( n ) a �− → a + e i Z( n ) = � i ≤ k φ i (Z( n − n i )) Christopher O’Neill (Texas A&M University)Computing the delta set and ω -primality in numerical monoidsJune 13, 2015 4 / 13

  9. A solution: dynamic programming Fix n ∈ S = � n 1 , . . . , n k � . For each i ≤ k , φ i : Z( n − n i ) − → Z( n ) a �− → a + e i Z( n ) = � i ≤ k φ i (Z( n − n i )) n ∈ S = � 6 , 9 , 20 � Z( n ) L( n ) 0 { 0 } { 0 } Christopher O’Neill (Texas A&M University)Computing the delta set and ω -primality in numerical monoidsJune 13, 2015 4 / 13

  10. A solution: dynamic programming Fix n ∈ S = � n 1 , . . . , n k � . For each i ≤ k , φ i : Z( n − n i ) − → Z( n ) a �− → a + e i Z( n ) = � i ≤ k φ i (Z( n − n i )) n ∈ S = � 6 , 9 , 20 � Z( n ) L( n ) 0 { 0 } { 0 } 6 6 0 � e 1 { e 1 } { 1 } Christopher O’Neill (Texas A&M University)Computing the delta set and ω -primality in numerical monoidsJune 13, 2015 4 / 13

  11. A solution: dynamic programming Fix n ∈ S = � n 1 , . . . , n k � . For each i ≤ k , φ i : Z( n − n i ) − → Z( n ) a �− → a + e i Z( n ) = � i ≤ k φ i (Z( n − n i )) n ∈ S = � 6 , 9 , 20 � Z( n ) L( n ) 0 { 0 } { 0 } 6 6 0 � e 1 { e 1 } { 1 } 9 9 0 � e 2 { e 2 } { 1 } Christopher O’Neill (Texas A&M University)Computing the delta set and ω -primality in numerical monoidsJune 13, 2015 4 / 13

  12. A solution: dynamic programming Fix n ∈ S = � n 1 , . . . , n k � . For each i ≤ k , φ i : Z( n − n i ) − → Z( n ) a �− → a + e i Z( n ) = � i ≤ k φ i (Z( n − n i )) n ∈ S = � 6 , 9 , 20 � Z( n ) L( n ) 0 { 0 } { 0 } 6 6 0 � e 1 { e 1 } { 1 } 9 9 0 � e 2 { e 2 } { 1 } 6 12 e 1 � 2 e 1 { 2 e 1 } { 2 } Christopher O’Neill (Texas A&M University)Computing the delta set and ω -primality in numerical monoidsJune 13, 2015 4 / 13

  13. A solution: dynamic programming Fix n ∈ S = � n 1 , . . . , n k � . For each i ≤ k , φ i : Z( n − n i ) − → Z( n ) a �− → a + e i Z( n ) = � i ≤ k φ i (Z( n − n i )) n ∈ S = � 6 , 9 , 20 � Z( n ) L( n ) 0 { 0 } { 0 } 6 6 0 � e 1 { e 1 } { 1 } 9 9 0 � e 2 { e 2 } { 1 } 6 12 e 1 � 2 e 1 { 2 e 1 } { 2 } 6 15 � (1 , 1 , 0) { (1 , 1 , 0) } { 2 } e 2 Christopher O’Neill (Texas A&M University)Computing the delta set and ω -primality in numerical monoidsJune 13, 2015 4 / 13

  14. A solution: dynamic programming Fix n ∈ S = � n 1 , . . . , n k � . For each i ≤ k , φ i : Z( n − n i ) − → Z( n ) a �− → a + e i Z( n ) = � i ≤ k φ i (Z( n − n i )) n ∈ S = � 6 , 9 , 20 � Z( n ) L( n ) 0 { 0 } { 0 } 6 6 0 � e 1 { e 1 } { 1 } 9 9 0 � e 2 { e 2 } { 1 } 6 12 e 1 � 2 e 1 { 2 e 1 } { 2 } 6 15 � (1 , 1 , 0) { (1 , 1 , 0) } { 2 } e 2 9 � (1 , 1 , 0) e 1 Christopher O’Neill (Texas A&M University)Computing the delta set and ω -primality in numerical monoidsJune 13, 2015 4 / 13

  15. A solution: dynamic programming Fix n ∈ S = � n 1 , . . . , n k � . For each i ≤ k , φ i : Z( n − n i ) − → Z( n ) a �− → a + e i Z( n ) = � i ≤ k φ i (Z( n − n i )) n ∈ S = � 6 , 9 , 20 � Z( n ) L( n ) 0 { 0 } { 0 } 6 6 0 � e 1 { e 1 } { 1 } 9 9 0 � e 2 { e 2 } { 1 } 6 12 e 1 � 2 e 1 { 2 e 1 } { 2 } 6 15 � (1 , 1 , 0) { (1 , 1 , 0) } { 2 } e 2 9 � (1 , 1 , 0) e 1 6 18 2 e 1 � 3 e 1 { 3 e 1 , 2 e 2 } { 2 , 3 } Christopher O’Neill (Texas A&M University)Computing the delta set and ω -primality in numerical monoidsJune 13, 2015 4 / 13

  16. A solution: dynamic programming Fix n ∈ S = � n 1 , . . . , n k � . For each i ≤ k , φ i : Z( n − n i ) − → Z( n ) a �− → a + e i Z( n ) = � i ≤ k φ i (Z( n − n i )) n ∈ S = � 6 , 9 , 20 � Z( n ) L( n ) 0 { 0 } { 0 } 6 6 0 � e 1 { e 1 } { 1 } 9 9 0 � e 2 { e 2 } { 1 } 6 12 e 1 � 2 e 1 { 2 e 1 } { 2 } 6 15 � (1 , 1 , 0) { (1 , 1 , 0) } { 2 } e 2 9 � (1 , 1 , 0) e 1 6 18 2 e 1 � 3 e 1 { 3 e 1 , 2 e 2 } { 2 , 3 } 9 � 2 e 2 e 2 Christopher O’Neill (Texas A&M University)Computing the delta set and ω -primality in numerical monoidsJune 13, 2015 4 / 13

  17. A solution: dynamic programming Fix n ∈ S = � n 1 , . . . , n k � . For each i ≤ k , φ i : Z( n − n i ) − → Z( n ) a �− → a + e i Z( n ) = � i ≤ k φ i (Z( n − n i )) n ∈ S = � 6 , 9 , 20 � Z( n ) L( n ) 0 { 0 } { 0 } 6 6 0 � e 1 { e 1 } { 1 } 9 9 0 � e 2 { e 2 } { 1 } 6 12 e 1 � 2 e 1 { 2 e 1 } { 2 } 6 15 � (1 , 1 , 0) { (1 , 1 , 0) } { 2 } e 2 9 � (1 , 1 , 0) e 1 6 18 2 e 1 � 3 e 1 { 3 e 1 , 2 e 2 } { 2 , 3 } 9 � 2 e 2 e 2 20 20 0 � e 3 { e 3 } { 1 } Christopher O’Neill (Texas A&M University)Computing the delta set and ω -primality in numerical monoidsJune 13, 2015 4 / 13

  18. A solution: dynamic programming Fix n ∈ S = � n 1 , . . . , n k � . For each i ≤ k , φ i : Z( n − n i ) − → Z( n ) a �− → a + e i Z( n ) = � i ≤ k φ i (Z( n − n i )) n ∈ S = � 6 , 9 , 20 � Z( n ) L( n ) 0 { 0 } { 0 } 6 6 0 � e 1 { e 1 } { 1 } 9 9 0 � e 2 { e 2 } { 1 } 6 12 e 1 � 2 e 1 { 2 e 1 } { 2 } 6 15 � (1 , 1 , 0) { (1 , 1 , 0) } { 2 } e 2 9 � (1 , 1 , 0) e 1 6 18 2 e 1 � 3 e 1 { 3 e 1 , 2 e 2 } { 2 , 3 } 9 � 2 e 2 e 2 20 20 0 � e 3 { e 3 } { 1 } . . . . . . . . . Christopher O’Neill (Texas A&M University)Computing the delta set and ω -primality in numerical monoidsJune 13, 2015 4 / 13

  19. A solution: dynamic programming Fix n ∈ S = � n 1 , . . . , n k � . For each i ≤ k , φ i : Z( n − n i ) − → Z( n ) ψ i : L( n − n i ) − → L( n ) a �− → a + e i Z( n ) = � i ≤ k φ i (Z( n − n i )) n ∈ S = � 6 , 9 , 20 � Z( n ) L( n ) 0 { 0 } { 0 } 6 6 0 � e 1 { e 1 } { 1 } 9 9 0 � e 2 { e 2 } { 1 } 6 12 e 1 � 2 e 1 { 2 e 1 } { 2 } 6 15 � (1 , 1 , 0) { (1 , 1 , 0) } { 2 } e 2 9 � (1 , 1 , 0) e 1 6 18 2 e 1 � 3 e 1 { 3 e 1 , 2 e 2 } { 2 , 3 } 9 � 2 e 2 e 2 20 20 0 � e 3 { e 3 } { 1 } . . . . . . . . . Christopher O’Neill (Texas A&M University)Computing the delta set and ω -primality in numerical monoidsJune 13, 2015 4 / 13

  20. A solution: dynamic programming Fix n ∈ S = � n 1 , . . . , n k � . For each i ≤ k , φ i : Z( n − n i ) − → Z( n ) ψ i : L( n − n i ) − → L( n ) a �− → a + e i ℓ �− → ℓ + 1 Z( n ) = � i ≤ k φ i (Z( n − n i )) n ∈ S = � 6 , 9 , 20 � Z( n ) L( n ) 0 { 0 } { 0 } 6 6 0 � e 1 { e 1 } { 1 } 9 9 0 � e 2 { e 2 } { 1 } 6 12 e 1 � 2 e 1 { 2 e 1 } { 2 } 6 15 � (1 , 1 , 0) { (1 , 1 , 0) } { 2 } e 2 9 � (1 , 1 , 0) e 1 6 18 2 e 1 � 3 e 1 { 3 e 1 , 2 e 2 } { 2 , 3 } 9 � 2 e 2 e 2 20 20 0 � e 3 { e 3 } { 1 } . . . . . . . . . Christopher O’Neill (Texas A&M University)Computing the delta set and ω -primality in numerical monoidsJune 13, 2015 4 / 13

  21. A solution: dynamic programming Fix n ∈ S = � n 1 , . . . , n k � . For each i ≤ k , φ i : Z( n − n i ) − → Z( n ) ψ i : L( n − n i ) − → L( n ) a �− → a + e i ℓ �− → ℓ + 1 Z( n ) = � i ≤ k φ i (Z( n − n i )) L( n ) = � i ≤ k ψ i (L( n − n i )) n ∈ S = � 6 , 9 , 20 � Z( n ) L( n ) 0 { 0 } { 0 } 6 6 0 � e 1 { e 1 } { 1 } 9 9 0 � e 2 { e 2 } { 1 } 6 12 e 1 � 2 e 1 { 2 e 1 } { 2 } 6 15 � (1 , 1 , 0) { (1 , 1 , 0) } { 2 } e 2 9 � (1 , 1 , 0) e 1 6 18 2 e 1 � 3 e 1 { 3 e 1 , 2 e 2 } { 2 , 3 } 9 � 2 e 2 e 2 20 20 0 � e 3 { e 3 } { 1 } . . . . . . . . . Christopher O’Neill (Texas A&M University)Computing the delta set and ω -primality in numerical monoidsJune 13, 2015 4 / 13

  22. A solution: dynamic programming Fix n ∈ S = � n 1 , . . . , n k � . For each i ≤ k , φ i : Z( n − n i ) − → Z( n ) ψ i : L( n − n i ) − → L( n ) a �− → a + e i ℓ �− → ℓ + 1 Z( n ) = � i ≤ k φ i (Z( n − n i )) L( n ) = � i ≤ k ψ i (L( n − n i )) n ∈ S = � 6 , 9 , 20 � L( n ) 0 { 0 } 6 9 12 15 18 20 . . . Christopher O’Neill (Texas A&M University)Computing the delta set and ω -primality in numerical monoidsJune 13, 2015 4 / 13

  23. A solution: dynamic programming Fix n ∈ S = � n 1 , . . . , n k � . For each i ≤ k , φ i : Z( n − n i ) − → Z( n ) ψ i : L( n − n i ) − → L( n ) a �− → a + e i ℓ �− → ℓ + 1 Z( n ) = � i ≤ k φ i (Z( n − n i )) L( n ) = � i ≤ k ψ i (L( n − n i )) n ∈ S = � 6 , 9 , 20 � L( n ) 0 { 0 } 6 6 { 1 } 0 � 1 9 12 15 18 20 . . . Christopher O’Neill (Texas A&M University)Computing the delta set and ω -primality in numerical monoidsJune 13, 2015 4 / 13

  24. A solution: dynamic programming Fix n ∈ S = � n 1 , . . . , n k � . For each i ≤ k , φ i : Z( n − n i ) − → Z( n ) ψ i : L( n − n i ) − → L( n ) a �− → a + e i ℓ �− → ℓ + 1 Z( n ) = � i ≤ k φ i (Z( n − n i )) L( n ) = � i ≤ k ψ i (L( n − n i )) n ∈ S = � 6 , 9 , 20 � L( n ) 0 { 0 } 6 6 { 1 } 0 � 1 9 9 { 1 } 0 � 1 12 15 18 20 . . . Christopher O’Neill (Texas A&M University)Computing the delta set and ω -primality in numerical monoidsJune 13, 2015 4 / 13

  25. A solution: dynamic programming Fix n ∈ S = � n 1 , . . . , n k � . For each i ≤ k , φ i : Z( n − n i ) − → Z( n ) ψ i : L( n − n i ) − → L( n ) a �− → a + e i ℓ �− → ℓ + 1 Z( n ) = � i ≤ k φ i (Z( n − n i )) L( n ) = � i ≤ k ψ i (L( n − n i )) n ∈ S = � 6 , 9 , 20 � L( n ) 0 { 0 } 6 6 { 1 } 0 � 1 9 9 { 1 } 0 � 1 6 12 { 2 } 1 � 2 15 18 20 . . . Christopher O’Neill (Texas A&M University)Computing the delta set and ω -primality in numerical monoidsJune 13, 2015 4 / 13

  26. A solution: dynamic programming Fix n ∈ S = � n 1 , . . . , n k � . For each i ≤ k , φ i : Z( n − n i ) − → Z( n ) ψ i : L( n − n i ) − → L( n ) a �− → a + e i ℓ �− → ℓ + 1 Z( n ) = � i ≤ k φ i (Z( n − n i )) L( n ) = � i ≤ k ψ i (L( n − n i )) n ∈ S = � 6 , 9 , 20 � L( n ) 0 { 0 } 6 6 { 1 } 0 � 1 9 9 { 1 } 0 � 1 6 12 { 2 } 1 � 2 6 15 { 2 } 1 � 2 18 20 . . . Christopher O’Neill (Texas A&M University)Computing the delta set and ω -primality in numerical monoidsJune 13, 2015 4 / 13

  27. A solution: dynamic programming Fix n ∈ S = � n 1 , . . . , n k � . For each i ≤ k , φ i : Z( n − n i ) − → Z( n ) ψ i : L( n − n i ) − → L( n ) a �− → a + e i ℓ �− → ℓ + 1 Z( n ) = � i ≤ k φ i (Z( n − n i )) L( n ) = � i ≤ k ψ i (L( n − n i )) n ∈ S = � 6 , 9 , 20 � L( n ) 0 { 0 } 6 6 { 1 } 0 � 1 9 9 { 1 } 0 � 1 6 12 { 2 } 1 � 2 6 15 { 2 } 1 � 2 9 1 � 2 18 20 . . . Christopher O’Neill (Texas A&M University)Computing the delta set and ω -primality in numerical monoidsJune 13, 2015 4 / 13

  28. A solution: dynamic programming Fix n ∈ S = � n 1 , . . . , n k � . For each i ≤ k , φ i : Z( n − n i ) − → Z( n ) ψ i : L( n − n i ) − → L( n ) a �− → a + e i ℓ �− → ℓ + 1 Z( n ) = � i ≤ k φ i (Z( n − n i )) L( n ) = � i ≤ k ψ i (L( n − n i )) n ∈ S = � 6 , 9 , 20 � L( n ) 0 { 0 } 6 6 { 1 } 0 � 1 9 9 { 1 } 0 � 1 6 12 { 2 } 1 � 2 6 15 { 2 } 1 � 2 9 1 � 2 6 18 { 2 , 3 } 2 � 3 20 . . . Christopher O’Neill (Texas A&M University)Computing the delta set and ω -primality in numerical monoidsJune 13, 2015 4 / 13

  29. A solution: dynamic programming Fix n ∈ S = � n 1 , . . . , n k � . For each i ≤ k , φ i : Z( n − n i ) − → Z( n ) ψ i : L( n − n i ) − → L( n ) a �− → a + e i ℓ �− → ℓ + 1 Z( n ) = � i ≤ k φ i (Z( n − n i )) L( n ) = � i ≤ k ψ i (L( n − n i )) n ∈ S = � 6 , 9 , 20 � L( n ) 0 { 0 } 6 6 { 1 } 0 � 1 9 9 { 1 } 0 � 1 6 12 { 2 } 1 � 2 6 15 { 2 } 1 � 2 9 1 � 2 6 18 { 2 , 3 } 2 � 3 9 1 � 2 20 . . . Christopher O’Neill (Texas A&M University)Computing the delta set and ω -primality in numerical monoidsJune 13, 2015 4 / 13

  30. A solution: dynamic programming Fix n ∈ S = � n 1 , . . . , n k � . For each i ≤ k , φ i : Z( n − n i ) − → Z( n ) ψ i : L( n − n i ) − → L( n ) a �− → a + e i ℓ �− → ℓ + 1 Z( n ) = � i ≤ k φ i (Z( n − n i )) L( n ) = � i ≤ k ψ i (L( n − n i )) n ∈ S = � 6 , 9 , 20 � L( n ) 0 { 0 } 6 6 { 1 } 0 � 1 9 9 { 1 } 0 � 1 6 12 { 2 } 1 � 2 6 15 { 2 } 1 � 2 9 1 � 2 6 18 { 2 , 3 } 2 � 3 9 1 � 2 20 20 { 1 } 0 � 1 . . . Christopher O’Neill (Texas A&M University)Computing the delta set and ω -primality in numerical monoidsJune 13, 2015 4 / 13

  31. A solution: dynamic programming Fix n ∈ S = � n 1 , . . . , n k � . For each i ≤ k , φ i : Z( n − n i ) − → Z( n ) ψ i : L( n − n i ) − → L( n ) a �− → a + e i ℓ �− → ℓ + 1 Z( n ) = � i ≤ k φ i (Z( n − n i )) L( n ) = � i ≤ k ψ i (L( n − n i )) n ∈ S = � 6 , 9 , 20 � L( n ) 0 { 0 } 6 6 { 1 } 0 � 1 9 9 { 1 } 0 � 1 6 12 { 2 } 1 � 2 6 15 { 2 } 1 � 2 9 1 � 2 6 18 { 2 , 3 } 2 � 3 9 1 � 2 20 20 { 1 } 0 � 1 . . . . . . . . . Christopher O’Neill (Texas A&M University)Computing the delta set and ω -primality in numerical monoidsJune 13, 2015 4 / 13

  32. Computing the delta set dynamically Theorem (Garc´ ıa-Garc´ ıa–Moreno-Fr´ ıas–Vigneron-Tenorio, 2014) S = � n 1 , . . . , n k � . For n ≥ N S , ∆( n ) = ∆( n + lcm( n 1 , n k )) . Christopher O’Neill (Texas A&M University)Computing the delta set and ω -primality in numerical monoidsJune 13, 2015 5 / 13

  33. Computing the delta set dynamically Theorem (Garc´ ıa-Garc´ ıa–Moreno-Fr´ ıas–Vigneron-Tenorio, 2014) S = � n 1 , . . . , n k � . For n ≥ N S , ∆( n ) = ∆( n + lcm( n 1 , n k )) . For n ∈ S with 0 ≤ n ≤ N S + lcm( n 1 , n k ), compute: Z( n ) = { a ∈ N k : n = a 1 n 1 + · · · + a k n k } Z( n ) � L( n ) L( n ) � ∆( n ) Compute ∆( S ) = � n ∆( n ). Christopher O’Neill (Texas A&M University)Computing the delta set and ω -primality in numerical monoidsJune 13, 2015 5 / 13

  34. Computing the delta set dynamically Theorem (Garc´ ıa-Garc´ ıa–Moreno-Fr´ ıas–Vigneron-Tenorio, 2014) S = � n 1 , . . . , n k � . For n ≥ N S , ∆( n ) = ∆( n + lcm( n 1 , n k )) . For n ∈ S with 0 ≤ n ≤ N S + lcm( n 1 , n k ), compute: Z( n ) = { a ∈ N k : n = a 1 n 1 + · · · + a k n k } L( n − ∗ ) � L( n ) L( n ) � ∆( n ) Compute ∆( S ) = � n ∆( n ). Christopher O’Neill (Texas A&M University)Computing the delta set and ω -primality in numerical monoidsJune 13, 2015 5 / 13

  35. Computing the delta set dynamically Theorem (Garc´ ıa-Garc´ ıa–Moreno-Fr´ ıas–Vigneron-Tenorio, 2014) S = � n 1 , . . . , n k � . For n ≥ N S , ∆( n ) = ∆( n + lcm( n 1 , n k )) . For n ∈ S with 0 ≤ n ≤ N S + lcm( n 1 , n k ), compute: Z( n ) = { a ∈ N k : n = a 1 n 1 + · · · + a k n k } L( n − ∗ ) � L( n ) L( n ) � ∆( n ) Compute ∆( S ) = � n ∆( n ). This is significantly faster! Christopher O’Neill (Texas A&M University)Computing the delta set and ω -primality in numerical monoidsJune 13, 2015 5 / 13

  36. Computing the delta set dynamically Theorem (Garc´ ıa-Garc´ ıa–Moreno-Fr´ ıas–Vigneron-Tenorio, 2014) S = � n 1 , . . . , n k � . For n ≥ N S , ∆( n ) = ∆( n + lcm( n 1 , n k )) . For n ∈ S with 0 ≤ n ≤ N S + lcm( n 1 , n k ), compute: Z( n ) = { a ∈ N k : n = a 1 n 1 + · · · + a k n k } L( n − ∗ ) � L( n ) L( n ) � ∆( n ) Compute ∆( S ) = � n ∆( n ). This is significantly faster! n k − 1 | Z( n ) | ≈ Christopher O’Neill (Texas A&M University)Computing the delta set and ω -primality in numerical monoidsJune 13, 2015 5 / 13

  37. Computing the delta set dynamically Theorem (Garc´ ıa-Garc´ ıa–Moreno-Fr´ ıas–Vigneron-Tenorio, 2014) S = � n 1 , . . . , n k � . For n ≥ N S , ∆( n ) = ∆( n + lcm( n 1 , n k )) . For n ∈ S with 0 ≤ n ≤ N S + lcm( n 1 , n k ), compute: Z( n ) = { a ∈ N k : n = a 1 n 1 + · · · + a k n k } L( n − ∗ ) � L( n ) L( n ) � ∆( n ) Compute ∆( S ) = � n ∆( n ). This is significantly faster! n k − 1 | Z( n ) | ≈ | L( n ) | ≈ n Christopher O’Neill (Texas A&M University)Computing the delta set and ω -primality in numerical monoidsJune 13, 2015 5 / 13

  38. Runtime comparison Christopher O’Neill (Texas A&M University)Computing the delta set and ω -primality in numerical monoidsJune 13, 2015 6 / 13

  39. Runtime comparison S N S ∆( S ) Existing Dynamic � 7 , 15 , 17 , 18 , 20 � 1935 { 1 , 2 , 3 } 1m 28s 146ms � 11 , 53 , 73 , 87 � 14381 { 2 , 4 , 6 , 8 , 10 , 22 } 0m 49s 2.5s � 31 , 73 , 77 , 87 , 91 � 31364 { 2 , 4 , 6 } 400m 12s 4.2s � 100 , 121 , 142 , 163 , 284 � 24850 { 21 } ——— 0m 3.6s � 1001 , 1211 , 1421 , 1631 , 2841 � 2063141 { 10 , 20 , 30 } ——— 1m 56s Christopher O’Neill (Texas A&M University)Computing the delta set and ω -primality in numerical monoidsJune 13, 2015 6 / 13

  40. Runtime comparison S N S ∆( S ) Existing Dynamic � 7 , 15 , 17 , 18 , 20 � 1935 { 1 , 2 , 3 } 1m 28s 146ms � 11 , 53 , 73 , 87 � 14381 { 2 , 4 , 6 , 8 , 10 , 22 } 0m 49s 2.5s � 31 , 73 , 77 , 87 , 91 � 31364 { 2 , 4 , 6 } 400m 12s 4.2s � 100 , 121 , 142 , 163 , 284 � 24850 { 21 } ——— 0m 3.6s � 1001 , 1211 , 1421 , 1631 , 2841 � 2063141 { 10 , 20 , 30 } ——— 1m 56s GAP Numerical Semigroups Package, available at http://www.gap-system.org/Packages/numericalsgps.html . Christopher O’Neill (Texas A&M University)Computing the delta set and ω -primality in numerical monoidsJune 13, 2015 6 / 13

  41. ω -primality As usual, n ∈ S = � n 1 , . . . , n k � . Christopher O’Neill (Texas A&M University)Computing the delta set and ω -primality in numerical monoidsJune 13, 2015 7 / 13

  42. ω -primality As usual, n ∈ S = � n 1 , . . . , n k � . Definition ( ω -primality) ω S ( n ) is the minimal m such that whenever ( � r i =1 x i ) − n ∈ S for r > m , there exists T ⊂ { 1 , . . . , r } with | T | ≤ m and ( � i ∈ T x i ) − n ∈ S . Christopher O’Neill (Texas A&M University)Computing the delta set and ω -primality in numerical monoidsJune 13, 2015 7 / 13

  43. ω -primality As usual, n ∈ S = � n 1 , . . . , n k � . Definition ( ω -primality) ω S ( n ) is the minimal m such that whenever ( � r i =1 x i ) − n ∈ S for r > m , there exists T ⊂ { 1 , . . . , r } with | T | ≤ m and ( � i ∈ T x i ) − n ∈ S . Definition A bullet for n ∈ S is a tuple b = ( b 1 , . . . , b k ) ∈ N k such that (i) b 1 n 1 + · · · + b k n k − n ∈ S , and (ii) b 1 n 1 + · · · + ( b i − 1) n i + · · · + b k n k − n / ∈ S for each b i > 0. The set of bullets of n is denoted bul( n ). Christopher O’Neill (Texas A&M University)Computing the delta set and ω -primality in numerical monoidsJune 13, 2015 7 / 13

  44. ω -primality As usual, n ∈ S = � n 1 , . . . , n k � . Definition ( ω -primality) ω S ( n ) is the minimal m such that whenever ( � r i =1 x i ) − n ∈ S for r > m , there exists T ⊂ { 1 , . . . , r } with | T | ≤ m and ( � i ∈ T x i ) − n ∈ S . Definition A bullet for n ∈ S is a tuple b = ( b 1 , . . . , b k ) ∈ N k such that (i) b 1 n 1 + · · · + b k n k − n ∈ S , and (ii) b 1 n 1 + · · · + ( b i − 1) n i + · · · + b k n k − n / ∈ S for each b i > 0. The set of bullets of n is denoted bul( n ). Proposition ω S ( n ) = max {| b | : b ∈ bul( n ) } . Christopher O’Neill (Texas A&M University)Computing the delta set and ω -primality in numerical monoidsJune 13, 2015 7 / 13

  45. Using bullets to compute ω -primality Algorithm: Compute bul( n ), then compute ω ( n ) = max {| b | : b ∈ bul( n ) } . Christopher O’Neill (Texas A&M University)Computing the delta set and ω -primality in numerical monoidsJune 13, 2015 8 / 13

  46. Using bullets to compute ω -primality Algorithm: Compute bul( n ), then compute ω ( n ) = max {| b | : b ∈ bul( n ) } . Example S = � 6 , 9 , 20 � = { 0 , 6 , 9 , 12 , 15 , 18 , 20 , 21 , . . . } . Christopher O’Neill (Texas A&M University)Computing the delta set and ω -primality in numerical monoidsJune 13, 2015 8 / 13

  47. Using bullets to compute ω -primality Algorithm: Compute bul( n ), then compute ω ( n ) = max {| b | : b ∈ bul( n ) } . Example S = � 6 , 9 , 20 � = { 0 , 6 , 9 , 12 , 15 , 18 , 20 , 21 , . . . } . “McNugget Monoid” Christopher O’Neill (Texas A&M University)Computing the delta set and ω -primality in numerical monoidsJune 13, 2015 8 / 13

  48. Using bullets to compute ω -primality Algorithm: Compute bul( n ), then compute ω ( n ) = max {| b | : b ∈ bul( n ) } . Example S = � 6 , 9 , 20 � = { 0 , 6 , 9 , 12 , 15 , 18 , 20 , 21 , . . . } . “McNugget Monoid” bul(60) = { (4 , 4 , 0) , (7 , 2 , 0) , (10 , 0 , 0) , (1 , 6 , 0) , (0 , 8 , 0) , (0 , 0 , 3) } Christopher O’Neill (Texas A&M University)Computing the delta set and ω -primality in numerical monoidsJune 13, 2015 8 / 13

  49. Using bullets to compute ω -primality Algorithm: Compute bul( n ), then compute ω ( n ) = max {| b | : b ∈ bul( n ) } . Example S = � 6 , 9 , 20 � = { 0 , 6 , 9 , 12 , 15 , 18 , 20 , 21 , . . . } . “McNugget Monoid” bul(60) = { (4 , 4 , 0) , (7 , 2 , 0) , (10 , 0 , 0) , (1 , 6 , 0) , (0 , 8 , 0) , (0 , 0 , 3) } 8 · 9 − 60 = 12 ∈ S Christopher O’Neill (Texas A&M University)Computing the delta set and ω -primality in numerical monoidsJune 13, 2015 8 / 13

  50. Using bullets to compute ω -primality Algorithm: Compute bul( n ), then compute ω ( n ) = max {| b | : b ∈ bul( n ) } . Example S = � 6 , 9 , 20 � = { 0 , 6 , 9 , 12 , 15 , 18 , 20 , 21 , . . . } . “McNugget Monoid” bul(60) = { (4 , 4 , 0) , (7 , 2 , 0) , (10 , 0 , 0) , (1 , 6 , 0) , (0 , 8 , 0) , (0 , 0 , 3) } 8 · 9 − 60 = 12 ∈ S 7 · 9 − 60 = 3 / ∈ S Christopher O’Neill (Texas A&M University)Computing the delta set and ω -primality in numerical monoidsJune 13, 2015 8 / 13

  51. Using bullets to compute ω -primality Algorithm: Compute bul( n ), then compute ω ( n ) = max {| b | : b ∈ bul( n ) } . Example S = � 6 , 9 , 20 � = { 0 , 6 , 9 , 12 , 15 , 18 , 20 , 21 , . . . } . “McNugget Monoid” bul(60) = { (4 , 4 , 0) , (7 , 2 , 0) , (10 , 0 , 0) , (1 , 6 , 0) , (0 , 8 , 0) , (0 , 0 , 3) } 8 · 9 − 60 = 12 ∈ S ⇒ (0 , 8 , 0) ∈ bul(60) 7 · 9 − 60 = 3 / ∈ S Christopher O’Neill (Texas A&M University)Computing the delta set and ω -primality in numerical monoidsJune 13, 2015 8 / 13

  52. Using bullets to compute ω -primality Algorithm: Compute bul( n ), then compute ω ( n ) = max {| b | : b ∈ bul( n ) } . Example S = � 6 , 9 , 20 � = { 0 , 6 , 9 , 12 , 15 , 18 , 20 , 21 , . . . } . “McNugget Monoid” bul(60) = { (4 , 4 , 0) , (7 , 2 , 0) , (10 , 0 , 0) , (1 , 6 , 0) , (0 , 8 , 0) , (0 , 0 , 3) } 8 · 9 − 60 = 12 ∈ S ⇒ (0 , 8 , 0) ∈ bul(60) 7 · 9 − 60 = 3 / ∈ S 1 · 6 + 6 · 9 − 60 = 0 ∈ S Christopher O’Neill (Texas A&M University)Computing the delta set and ω -primality in numerical monoidsJune 13, 2015 8 / 13

  53. Using bullets to compute ω -primality Algorithm: Compute bul( n ), then compute ω ( n ) = max {| b | : b ∈ bul( n ) } . Example S = � 6 , 9 , 20 � = { 0 , 6 , 9 , 12 , 15 , 18 , 20 , 21 , . . . } . “McNugget Monoid” bul(60) = { (4 , 4 , 0) , (7 , 2 , 0) , (10 , 0 , 0) , (1 , 6 , 0) , (0 , 8 , 0) , (0 , 0 , 3) } 8 · 9 − 60 = 12 ∈ S ⇒ (0 , 8 , 0) ∈ bul(60) 7 · 9 − 60 = 3 / ∈ S 1 · 6 + 6 · 9 − 60 = 0 ∈ S 6 · 9 − 60 = − 6 / ∈ S Christopher O’Neill (Texas A&M University)Computing the delta set and ω -primality in numerical monoidsJune 13, 2015 8 / 13

  54. Using bullets to compute ω -primality Algorithm: Compute bul( n ), then compute ω ( n ) = max {| b | : b ∈ bul( n ) } . Example S = � 6 , 9 , 20 � = { 0 , 6 , 9 , 12 , 15 , 18 , 20 , 21 , . . . } . “McNugget Monoid” bul(60) = { (4 , 4 , 0) , (7 , 2 , 0) , (10 , 0 , 0) , (1 , 6 , 0) , (0 , 8 , 0) , (0 , 0 , 3) } 8 · 9 − 60 = 12 ∈ S ⇒ (0 , 8 , 0) ∈ bul(60) 7 · 9 − 60 = 3 / ∈ S 1 · 6 + 6 · 9 − 60 = 0 ∈ S 6 · 9 − 60 = − 6 / ∈ S 1 · 6 + 5 · 9 − 60 = − 9 / ∈ S Christopher O’Neill (Texas A&M University)Computing the delta set and ω -primality in numerical monoidsJune 13, 2015 8 / 13

  55. Using bullets to compute ω -primality Algorithm: Compute bul( n ), then compute ω ( n ) = max {| b | : b ∈ bul( n ) } . Example S = � 6 , 9 , 20 � = { 0 , 6 , 9 , 12 , 15 , 18 , 20 , 21 , . . . } . “McNugget Monoid” bul(60) = { (4 , 4 , 0) , (7 , 2 , 0) , (10 , 0 , 0) , (1 , 6 , 0) , (0 , 8 , 0) , (0 , 0 , 3) } 8 · 9 − 60 = 12 ∈ S ⇒ (0 , 8 , 0) ∈ bul(60) 7 · 9 − 60 = 3 / ∈ S 1 · 6 + 6 · 9 − 60 = 0 ∈ S 6 · 9 − 60 = − 6 / ∈ S ⇒ (1 , 6 , 0) ∈ bul(60) 1 · 6 + 5 · 9 − 60 = − 9 / ∈ S Christopher O’Neill (Texas A&M University)Computing the delta set and ω -primality in numerical monoidsJune 13, 2015 8 / 13

  56. Using bullets to compute ω -primality Algorithm: Compute bul( n ), then compute ω ( n ) = max {| b | : b ∈ bul( n ) } . Example S = � 6 , 9 , 20 � = { 0 , 6 , 9 , 12 , 15 , 18 , 20 , 21 , . . . } . “McNugget Monoid” bul(60) = { (4 , 4 , 0) , (7 , 2 , 0) , (10 , 0 , 0) , (1 , 6 , 0) , (0 , 8 , 0) , (0 , 0 , 3) } 8 · 9 − 60 = 12 ∈ S ⇒ (0 , 8 , 0) ∈ bul(60) 7 · 9 − 60 = 3 / ∈ S 1 · 6 + 6 · 9 − 60 = 0 ∈ S 6 · 9 − 60 = − 6 / ∈ S ⇒ (1 , 6 , 0) ∈ bul(60) 1 · 6 + 5 · 9 − 60 = − 9 / ∈ S n ∈ S ω ( n ) mbul n ∈ S ω ( n ) mbul n ∈ S ω ( n ) mbul 6 3 3 e 3 15 4 4 e 1 21 5 5 e 1 9 3 3 e 3 18 3 3 e 1 24 4 4 e 1 12 3 3 e 3 20 10 10 e 1 26 11 11 e 1 Christopher O’Neill (Texas A&M University)Computing the delta set and ω -primality in numerical monoidsJune 13, 2015 8 / 13

  57. Using bullets to compute ω -primality Algorithm: Compute bul( n ), then compute ω ( n ) = max {| b | : b ∈ bul( n ) } . Example S = � 6 , 9 , 20 � = { 0 , 6 , 9 , 12 , 15 , 18 , 20 , 21 , . . . } . “McNugget Monoid” bul(60) = { (4 , 4 , 0) , (7 , 2 , 0) , (10 , 0 , 0) , (1 , 6 , 0) , (0 , 8 , 0) , (0 , 0 , 3) } 8 · 9 − 60 = 12 ∈ S ⇒ (0 , 8 , 0) ∈ bul(60) 7 · 9 − 60 = 3 / ∈ S 1 · 6 + 6 · 9 − 60 = 0 ∈ S 6 · 9 − 60 = − 6 / ∈ S ⇒ (1 , 6 , 0) ∈ bul(60) 1 · 6 + 5 · 9 − 60 = − 9 / ∈ S n ∈ S ω ( n ) mbul n ∈ S ω ( n ) mbul n ∈ S ω ( n ) mbul 6 3 3 e 3 15 4 4 e 1 21 5 5 e 1 9 3 3 e 3 18 3 3 e 1 24 4 4 e 1 12 3 3 e 3 20 10 10 e 1 26 11 11 e 1 Moral of this talk: bullets behave like factorizations! Christopher O’Neill (Texas A&M University)Computing the delta set and ω -primality in numerical monoidsJune 13, 2015 8 / 13

  58. Toward a dynamic algorithm. . . the inductive step Recall: for n ∈ S = � n 1 , . . . n k � , Z( n ) = { a ∈ N k : � k i =1 a i n i = n } . Christopher O’Neill (Texas A&M University)Computing the delta set and ω -primality in numerical monoidsJune 13, 2015 9 / 13

  59. Toward a dynamic algorithm. . . the inductive step Recall: for n ∈ S = � n 1 , . . . n k � , Z( n ) = { a ∈ N k : � k i =1 a i n i = n } . For each i ≤ k , Christopher O’Neill (Texas A&M University)Computing the delta set and ω -primality in numerical monoidsJune 13, 2015 9 / 13

  60. Toward a dynamic algorithm. . . the inductive step Recall: for n ∈ S = � n 1 , . . . n k � , Z( n ) = { a ∈ N k : � k i =1 a i n i = n } . For each i ≤ k , φ i : Z( n − n i ) − → Z( n ) �− → a + e i . a Christopher O’Neill (Texas A&M University)Computing the delta set and ω -primality in numerical monoidsJune 13, 2015 9 / 13

  61. Toward a dynamic algorithm. . . the inductive step Recall: for n ∈ S = � n 1 , . . . n k � , Z( n ) = { a ∈ N k : � k i =1 a i n i = n } . For each i ≤ k , φ i : Z( n − n i ) − → Z( n ) �− → a + e i . a In particular, � Z( n ) = φ i (Z( n − n i )) i ≤ k Christopher O’Neill (Texas A&M University)Computing the delta set and ω -primality in numerical monoidsJune 13, 2015 9 / 13

  62. Toward a dynamic algorithm. . . the inductive step Recall: for n ∈ S = � n 1 , . . . n k � , Z( n ) = { a ∈ N k : � k i =1 a i n i = n } . For each i ≤ k , φ i : Z( n − n i ) − → Z( n ) �− → a + e i . a In particular, � Z( n ) = φ i (Z( n − n i )) i ≤ k Definition/Proposition (Cover morphisms) Fix n ∈ S and i ≤ k . The i-th cover morphism for n is the map ψ i : bul( n − n i ) − → bul( n ) Christopher O’Neill (Texas A&M University)Computing the delta set and ω -primality in numerical monoidsJune 13, 2015 9 / 13

  63. Toward a dynamic algorithm. . . the inductive step Recall: for n ∈ S = � n 1 , . . . n k � , Z( n ) = { a ∈ N k : � k i =1 a i n i = n } . For each i ≤ k , φ i : Z( n − n i ) − → Z( n ) �− → a + e i . a In particular, � Z( n ) = φ i (Z( n − n i )) i ≤ k Definition/Proposition (Cover morphisms) Fix n ∈ S and i ≤ k . The i-th cover morphism for n is the map ψ i : bul( n − n i ) − → bul( n ) given by � � k b + e i j =1 b j n j − n − n i / ∈ S b �− → � k b j =1 b j n j − n − n i ∈ S Christopher O’Neill (Texas A&M University)Computing the delta set and ω -primality in numerical monoidsJune 13, 2015 9 / 13

  64. Toward a dynamic algorithm. . . the inductive step Recall: for n ∈ S = � n 1 , . . . n k � , Z( n ) = { a ∈ N k : � k i =1 a i n i = n } . For each i ≤ k , φ i : Z( n − n i ) − → Z( n ) �− → a + e i . a In particular, � Z( n ) = φ i (Z( n − n i )) i ≤ k Definition/Proposition (Cover morphisms) Fix n ∈ S and i ≤ k . The i-th cover morphism for n is the map ψ i : bul( n − n i ) − → bul( n ) given by � � k b + e i j =1 b j n j − n − n i / ∈ S b �− → � k b j =1 b j n j − n − n i ∈ S Moreover, bul( n ) = � i ≤ k ψ i (bul( n − n i )).** Christopher O’Neill (Texas A&M University)Computing the delta set and ω -primality in numerical monoidsJune 13, 2015 9 / 13

  65. Toward a dynamic algorithm. . . the base case Definition ( ω -primality in numerical monoids) Fix a numerical monoid S and n ∈ S . Christopher O’Neill (Texas A&M University)Computing the delta set and ω -primality in numerical monoids June 13, 2015 10 / 13

  66. Toward a dynamic algorithm. . . the base case Definition ( ω -primality in numerical monoids) Fix a numerical monoid S and n ∈ S . ω S ( n ) is the minimal m such that whenever ( � r i =1 x i ) − n ∈ S for r > m , there exists T ⊂ { 1 , . . . , r } with | T | ≤ m and ( � i ∈ T x i ) − n ∈ S . Christopher O’Neill (Texas A&M University)Computing the delta set and ω -primality in numerical monoids June 13, 2015 10 / 13

  67. Toward a dynamic algorithm. . . the base case Definition ( ω -primality in numerical monoids) Fix a numerical monoid S and n ∈ S . ω S ( n ) is the minimal m such that whenever ( � r i =1 x i ) − n ∈ S for r > m , there exists T ⊂ { 1 , . . . , r } with | T | ≤ m and ( � i ∈ T x i ) − n ∈ S . Christopher O’Neill (Texas A&M University)Computing the delta set and ω -primality in numerical monoids June 13, 2015 10 / 13

  68. Toward a dynamic algorithm. . . the base case Definition ( ω -primality in numerical monoids) Fix a numerical monoid S and n ∈ Z = q( S ). ω S ( n ) is the minimal m such that whenever ( � r i =1 x i ) − n ∈ S for r > m , there exists T ⊂ { 1 , . . . , r } with | T | ≤ m and ( � i ∈ T x i ) − n ∈ S . Christopher O’Neill (Texas A&M University)Computing the delta set and ω -primality in numerical monoids June 13, 2015 10 / 13

  69. Toward a dynamic algorithm. . . the base case Definition ( ω -primality in numerical monoids) Fix a numerical monoid S and n ∈ Z = q( S ). ω S ( n ) is the minimal m such that whenever ( � r i =1 x i ) − n ∈ S for r > m , there exists T ⊂ { 1 , . . . , r } with | T | ≤ m and ( � i ∈ T x i ) − n ∈ S . Remark All properties of ω extend from S to Z . Christopher O’Neill (Texas A&M University)Computing the delta set and ω -primality in numerical monoids June 13, 2015 10 / 13

  70. Toward a dynamic algorithm. . . the base case Definition ( ω -primality in numerical monoids) Fix a numerical monoid S and n ∈ Z = q( S ). ω S ( n ) is the minimal m such that whenever ( � r i =1 x i ) − n ∈ S for r > m , there exists T ⊂ { 1 , . . . , r } with | T | ≤ m and ( � i ∈ T x i ) − n ∈ S . Remark All properties of ω extend from S to Z . Proposition For n ∈ Z , the following are equivalent: (i) ω ( n ) = 0, (ii) bul( n ) = { 0 } , (iii) − n ∈ S . Christopher O’Neill (Texas A&M University)Computing the delta set and ω -primality in numerical monoids June 13, 2015 10 / 13

  71. A dynamic algorithm! Example S = � 6 , 9 , 20 � = { 0 , 6 , 9 , 12 , 15 , 18 , 20 , 21 , . . . } . Christopher O’Neill (Texas A&M University)Computing the delta set and ω -primality in numerical monoids June 13, 2015 11 / 13

  72. A dynamic algorithm! Example S = � 6 , 9 , 20 � = { 0 , 6 , 9 , 12 , 15 , 18 , 20 , 21 , . . . } . n ∈ Z ω ( n ) bul( n ) n ∈ Z ω ( n ) bul( n ) Christopher O’Neill (Texas A&M University)Computing the delta set and ω -primality in numerical monoids June 13, 2015 11 / 13

  73. A dynamic algorithm! Example S = � 6 , 9 , 20 � = { 0 , 6 , 9 , 12 , 15 , 18 , 20 , 21 , . . . } . n ∈ Z ω ( n ) bul( n ) n ∈ Z ω ( n ) bul( n ) ≤ − 44 0 { 0 } Christopher O’Neill (Texas A&M University)Computing the delta set and ω -primality in numerical monoids June 13, 2015 11 / 13

  74. A dynamic algorithm! Example S = � 6 , 9 , 20 � = { 0 , 6 , 9 , 12 , 15 , 18 , 20 , 21 , . . . } . n ∈ Z ω ( n ) bul( n ) n ∈ Z ω ( n ) bul( n ) ≤ − 44 0 { 0 } − 43 1 { e 1 , e 2 , e 3 } Christopher O’Neill (Texas A&M University)Computing the delta set and ω -primality in numerical monoids June 13, 2015 11 / 13

  75. A dynamic algorithm! Example S = � 6 , 9 , 20 � = { 0 , 6 , 9 , 12 , 15 , 18 , 20 , 21 , . . . } . n ∈ Z ω ( n ) bul( n ) n ∈ Z ω ( n ) bul( n ) ≤ − 44 0 { 0 } − 43 1 { e 1 , e 2 , e 3 } − 42 0 { 0 } Christopher O’Neill (Texas A&M University)Computing the delta set and ω -primality in numerical monoids June 13, 2015 11 / 13

  76. A dynamic algorithm! Example S = � 6 , 9 , 20 � = { 0 , 6 , 9 , 12 , 15 , 18 , 20 , 21 , . . . } . n ∈ Z ω ( n ) bul( n ) n ∈ Z ω ( n ) bul( n ) ≤ − 44 0 { 0 } − 43 1 { e 1 , e 2 , e 3 } − 42 0 { 0 } . . . . . . . . . − 38 0 { 0 } Christopher O’Neill (Texas A&M University)Computing the delta set and ω -primality in numerical monoids June 13, 2015 11 / 13

  77. A dynamic algorithm! Example S = � 6 , 9 , 20 � = { 0 , 6 , 9 , 12 , 15 , 18 , 20 , 21 , . . . } . n ∈ Z ω ( n ) bul( n ) n ∈ Z ω ( n ) bul( n ) ≤ − 44 0 { 0 } − 43 1 { e 1 , e 2 , e 3 } − 42 0 { 0 } . . . . . . . . . − 38 0 { 0 } − 37 2 { 2 e 1 , e 2 , e 3 } Christopher O’Neill (Texas A&M University)Computing the delta set and ω -primality in numerical monoids June 13, 2015 11 / 13

  78. A dynamic algorithm! Example S = � 6 , 9 , 20 � = { 0 , 6 , 9 , 12 , 15 , 18 , 20 , 21 , . . . } . n ∈ Z ω ( n ) bul( n ) n ∈ Z ω ( n ) bul( n ) ≤ − 44 0 { 0 } − 43 1 { e 1 , e 2 , e 3 } − 42 0 { 0 } . . . . . . . . . − 38 0 { 0 } − 37 2 { 2 e 1 , e 2 , e 3 } − 36 0 { 0 } − 35 0 { 0 } Christopher O’Neill (Texas A&M University)Computing the delta set and ω -primality in numerical monoids June 13, 2015 11 / 13

  79. A dynamic algorithm! Example S = � 6 , 9 , 20 � = { 0 , 6 , 9 , 12 , 15 , 18 , 20 , 21 , . . . } . n ∈ Z ω ( n ) bul( n ) n ∈ Z ω ( n ) bul( n ) ≤ − 44 0 { 0 } − 43 1 { e 1 , e 2 , e 3 } − 42 0 { 0 } . . . . . . . . . − 38 0 { 0 } − 37 2 { 2 e 1 , e 2 , e 3 } − 36 0 { 0 } − 35 0 { 0 } − 34 2 { e 1 , 2 e 2 , e 3 } Christopher O’Neill (Texas A&M University)Computing the delta set and ω -primality in numerical monoids June 13, 2015 11 / 13

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