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Shifting numerical monoids Christopher ONeill University of - PowerPoint PPT Presentation

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Shifting numerical monoids Christopher ONeill University of California Davis coneill@math.ucdavis.edu Joint with Rebecca Conaway*, Felix Gotti, Jesse Horton*, Roberto Pelayo, Mesa Williams*, and Brian Wissman * = undergraduate student

  1. To shift a numerical monoid. . . Fix S = � r 1 , . . . , r k � ⊂ ( N , +) , and let M n = � n , n + r 1 , . . . , n + r k � . Delta set ∆( M n ) : successive factorization length differences in M n . Theorem (Chapman-Kaplan-Lemburg-Niles-Zlogar, 2014) The delta set ∆( M n ) is singleton for n ≫ 0 . M n = � n , n + 6 , n + 9 , n + 20 � : Christopher O’Neill (UC Davis) Shifting numerical monoids March 21, 2017 4 / 25

  2. To shift a numerical monoid. . . Fix S = � r 1 , . . . , r k � ⊂ ( N , +) , and let M n = � n , n + r 1 , . . . , n + r k � . Delta set ∆( M n ) : successive factorization length differences in M n . Theorem (Chapman-Kaplan-Lemburg-Niles-Zlogar, 2014) The delta set ∆( M n ) is singleton for n ≫ 0 . M n = � n , n + 6 , n + 9 , n + 20 � : ∆( M n ) = { 1 } for all n ≥ 48 Christopher O’Neill (UC Davis) Shifting numerical monoids March 21, 2017 4 / 25

  3. 4 10 14 50 100 150 200 0 2 12 6 8 To shift a numerical monoid. . . Fix S = � r 1 , . . . , r k � ⊂ ( N , +) , and let M n = � n , n + r 1 , . . . , n + r k � . Christopher O’Neill (UC Davis) Shifting numerical monoids March 21, 2017 5 / 25

  4. 4 8 14 12 50 100 150 200 0 2 10 6 To shift a numerical monoid. . . Fix S = � r 1 , . . . , r k � ⊂ ( N , +) , and let M n = � n , n + r 1 , . . . , n + r k � . Catenary degree c ( M n ) : measures spread of factorizations in M n . Christopher O’Neill (UC Davis) Shifting numerical monoids March 21, 2017 5 / 25

  5. 2 6 14 12 10 50 100 150 200 0 8 4 To shift a numerical monoid. . . Fix S = � r 1 , . . . , r k � ⊂ ( N , +) , and let M n = � n , n + r 1 , . . . , n + r k � . Catenary degree c ( M n ) : measures spread of factorizations in M n . M n = � n , n + 6 , n + 9 , n + 20 � : Christopher O’Neill (UC Davis) Shifting numerical monoids March 21, 2017 5 / 25

  6. 2 6 14 12 10 50 100 150 200 0 8 4 To shift a numerical monoid. . . Fix S = � r 1 , . . . , r k � ⊂ ( N , +) , and let M n = � n , n + r 1 , . . . , n + r k � . Catenary degree c ( M n ) : measures spread of factorizations in M n . M n = � n , n + 6 , n + 9 , n + 20 � : Christopher O’Neill (UC Davis) Shifting numerical monoids March 21, 2017 5 / 25

  7. 2 4 14 12 10 8 50 100 150 200 0 6 To shift a numerical monoid. . . Fix S = � r 1 , . . . , r k � ⊂ ( N , +) , and let M n = � n , n + r 1 , . . . , n + r k � . Catenary degree c ( M n ) : measures spread of factorizations in M n . M n = � n , n + 6 , n + 9 , n + 20 � : c ( M n ) is periodic-linear (quasilinear) for n ≥ 126. Christopher O’Neill (UC Davis) Shifting numerical monoids March 21, 2017 5 / 25

  8. 1000 2000 3000 50 100 150 200 0 500 2500 1500 To shift a numerical monoid. . . Fix S = � r 1 , . . . , r k � ⊂ ( N , +) , and let M n = � n , n + r 1 , . . . , n + r k � . Christopher O’Neill (UC Davis) Shifting numerical monoids March 21, 2017 6 / 25

  9. 500 1500 3000 2500 50 100 150 200 0 2000 1000 To shift a numerical monoid. . . Fix S = � r 1 , . . . , r k � ⊂ ( N , +) , and let M n = � n , n + r 1 , . . . , n + r k � . Betti numbers β i ( M n ) : Betti numbers of the defining toric ideal I M n . Christopher O’Neill (UC Davis) Shifting numerical monoids March 21, 2017 6 / 25

  10. 0 500 3000 2500 2000 1500 50 100 150 200 1000 To shift a numerical monoid. . . Fix S = � r 1 , . . . , r k � ⊂ ( N , +) , and let M n = � n , n + r 1 , . . . , n + r k � . Betti numbers β i ( M n ) : Betti numbers of the defining toric ideal I M n . Theorem (Vu, 2014) The Betti numbers of M n are eventually r k -periodic in n. Christopher O’Neill (UC Davis) Shifting numerical monoids March 21, 2017 6 / 25

  11. 0 500 3000 2500 2000 1500 1000 50 100 150 200 To shift a numerical monoid. . . Fix S = � r 1 , . . . , r k � ⊂ ( N , +) , and let M n = � n , n + r 1 , . . . , n + r k � . Betti numbers β i ( M n ) : Betti numbers of the defining toric ideal I M n . Theorem (Vu, 2014) The Betti numbers of M n are eventually r k -periodic in n. M n = � n , n + 6 , n + 9 , n + 20 � : Graded degrees for β 0 ( M n ) Christopher O’Neill (UC Davis) Shifting numerical monoids March 21, 2017 6 / 25

  12. To shift a numerical monoid. . . Fix S = � r 1 , . . . , r k � ⊂ ( N , +) , and let M n = � n , n + r 1 , . . . , n + r k � . Christopher O’Neill (UC Davis) Shifting numerical monoids March 21, 2017 7 / 25

  13. To shift a numerical monoid. . . Fix S = � r 1 , . . . , r k � ⊂ ( N , +) , and let M n = � n , n + r 1 , . . . , n + r k � . Observations: Christopher O’Neill (UC Davis) Shifting numerical monoids March 21, 2017 7 / 25

  14. To shift a numerical monoid. . . Fix S = � r 1 , . . . , r k � ⊂ ( N , +) , and let M n = � n , n + r 1 , . . . , n + r k � . Observations: Known: the Betti numbers n �→ β i ( M n ) are eventually r k -periodic. Christopher O’Neill (UC Davis) Shifting numerical monoids March 21, 2017 7 / 25

  15. To shift a numerical monoid. . . Fix S = � r 1 , . . . , r k � ⊂ ( N , +) , and let M n = � n , n + r 1 , . . . , n + r k � . Observations: Known: the Betti numbers n �→ β i ( M n ) are eventually r k -periodic. Known: the function n �→ ∆( M n ) is eventually singleton. Christopher O’Neill (UC Davis) Shifting numerical monoids March 21, 2017 7 / 25

  16. To shift a numerical monoid. . . Fix S = � r 1 , . . . , r k � ⊂ ( N , +) , and let M n = � n , n + r 1 , . . . , n + r k � . Observations: Known: the Betti numbers n �→ β i ( M n ) are eventually r k -periodic. Known: the function n �→ ∆( M n ) is eventually singleton. Observed: the function n �→ c ( M n ) is eventually r k -quasilinear. Christopher O’Neill (UC Davis) Shifting numerical monoids March 21, 2017 7 / 25

  17. To shift a numerical monoid. . . Fix S = � r 1 , . . . , r k � ⊂ ( N , +) , and let M n = � n , n + r 1 , . . . , n + r k � . Observations: Known: the Betti numbers n �→ β i ( M n ) are eventually r k -periodic. Known: the function n �→ ∆( M n ) is eventually singleton. Observed: the function n �→ c ( M n ) is eventually r k -quasilinear. Underlying cause: Christopher O’Neill (UC Davis) Shifting numerical monoids March 21, 2017 7 / 25

  18. To shift a numerical monoid. . . Fix S = � r 1 , . . . , r k � ⊂ ( N , +) , and let M n = � n , n + r 1 , . . . , n + r k � . Observations: Known: the Betti numbers n �→ β i ( M n ) are eventually r k -periodic. Known: the function n �→ ∆( M n ) is eventually singleton. Observed: the function n �→ c ( M n ) is eventually r k -quasilinear. Underlying cause: minimal presentations! Christopher O’Neill (UC Davis) Shifting numerical monoids March 21, 2017 7 / 25

  19. Kernel congruences and minimal presentations Let S = � r 1 , . . . , r k � . a = ( a 1 , . . . , a k ) ∈ N k n = a 1 r 1 + · · · + a k r k � Christopher O’Neill (UC Davis) Shifting numerical monoids March 21, 2017 8 / 25

  20. Kernel congruences and minimal presentations Let S = � r 1 , . . . , r k � . a = ( a 1 , . . . , a k ) ∈ N k n = a 1 r 1 + · · · + a k r k � Factorization homomorphism: π : N k − → � r 1 , . . . , r k � a �− → a 1 r 1 + · · · + a k r k Christopher O’Neill (UC Davis) Shifting numerical monoids March 21, 2017 8 / 25

  21. Kernel congruences and minimal presentations Let S = � r 1 , . . . , r k � . a = ( a 1 , . . . , a k ) ∈ N k n = a 1 r 1 + · · · + a k r k � Factorization homomorphism: π : N k − → � r 1 , . . . , r k � a �− → a 1 r 1 + · · · + a k r k Definition The kernel ker π is the relation ∼ on N k with a ∼ b whenever π ( a ) = π ( b ) Christopher O’Neill (UC Davis) Shifting numerical monoids March 21, 2017 8 / 25

  22. Kernel congruences and minimal presentations Let S = � r 1 , . . . , r k � . a = ( a 1 , . . . , a k ) ∈ N k n = a 1 r 1 + · · · + a k r k � Factorization homomorphism: π : N k − → � r 1 , . . . , r k � a �− → a 1 r 1 + · · · + a k r k Definition The kernel ker π is the relation ∼ on N k with a ∼ b whenever π ( a ) = π ( b ) ker π is a congruence : an equivalence relation a ∼ a a ∼ b ⇒ b ∼ a a ∼ b and b ∼ c ⇒ a ∼ c Christopher O’Neill (UC Davis) Shifting numerical monoids March 21, 2017 8 / 25

  23. Kernel congruences and minimal presentations Let S = � r 1 , . . . , r k � . a = ( a 1 , . . . , a k ) ∈ N k n = a 1 r 1 + · · · + a k r k � Factorization homomorphism: π : N k − → � r 1 , . . . , r k � a �− → a 1 r 1 + · · · + a k r k Definition The kernel ker π is the relation ∼ on N k with a ∼ b whenever π ( a ) = π ( b ) ker π is a congruence : an equivalence relation a ∼ a a ∼ b ⇒ b ∼ a a ∼ b and b ∼ c ⇒ a ∼ c that is closed under translation . a ∼ b ⇒ a + c ∼ b + c Christopher O’Neill (UC Davis) Shifting numerical monoids March 21, 2017 8 / 25

  24. Kernel congruences and minimal presentations Let S = � r 1 , . . . , r k � . a = ( a 1 , . . . , a k ) ∈ N k n = a 1 r 1 + · · · + a k r k � Factorization homomorphism: π : N k − → � r 1 , . . . , r k � a �− → a 1 r 1 + · · · + a k r k Definition The kernel ker π is the relation ∼ on N k with a ∼ b whenever π ( a ) = π ( b ) ker π is a congruence : an equivalence relation a ∼ a a ∼ b ⇒ b ∼ a a ∼ b and b ∼ c ⇒ a ∼ c that is closed under translation . a ∼ b ⇒ a + c ∼ b + c Christopher O’Neill (UC Davis) Shifting numerical monoids March 21, 2017 9 / 25

  25. Kernel congruences and minimal presentations Let S = � r 1 , . . . , r k � . a = ( a 1 , . . . , a k ) ∈ N k n = a 1 r 1 + · · · + a k r k � Factorization homomorphism: Monomial map: π : N k − → � r 1 , . . . , r k � a �− → a 1 r 1 + · · · + a k r k Definition The kernel ker π is the relation ∼ on N k with a ∼ b whenever π ( a ) = π ( b ) ker π is a congruence : an equivalence relation a ∼ a a ∼ b ⇒ b ∼ a a ∼ b and b ∼ c ⇒ a ∼ c that is closed under translation . a ∼ b ⇒ a + c ∼ b + c Christopher O’Neill (UC Davis) Shifting numerical monoids March 21, 2017 9 / 25

  26. Kernel congruences and minimal presentations Let S = � r 1 , . . . , r k � . a = ( a 1 , . . . , a k ) ∈ N k n = a 1 r 1 + · · · + a k r k � Factorization homomorphism: Monomial map: π : N k − → � r 1 , . . . , r k � ϕ : k [ x 1 , . . . , x k ] − → k [ y ] y r i a �− → a 1 r 1 + · · · + a k r k x i �− → Definition The kernel ker π is the relation ∼ on N k with a ∼ b whenever π ( a ) = π ( b ) ker π is a congruence : an equivalence relation a ∼ a a ∼ b ⇒ b ∼ a a ∼ b and b ∼ c ⇒ a ∼ c that is closed under translation . a ∼ b ⇒ a + c ∼ b + c Christopher O’Neill (UC Davis) Shifting numerical monoids March 21, 2017 9 / 25

  27. Kernel congruences and minimal presentations Let S = � r 1 , . . . , r k � . a = ( a 1 , . . . , a k ) ∈ N k n = a 1 r 1 + · · · + a k r k � Factorization homomorphism: Monomial map: π : N k − → � r 1 , . . . , r k � ϕ : k [ x 1 , . . . , x k ] − → k [ y ] y r i a �− → a 1 r 1 + · · · + a k r k x i �− → Definition The kernel ker π is the relation ∼ on N k with a ∼ b whenever x a − x b ∈ I S = ker ϕ π ( a ) = π ( b ) ker π is a congruence : an equivalence relation a ∼ a a ∼ b ⇒ b ∼ a a ∼ b and b ∼ c ⇒ a ∼ c that is closed under translation . a ∼ b ⇒ a + c ∼ b + c Christopher O’Neill (UC Davis) Shifting numerical monoids March 21, 2017 9 / 25

  28. Kernel congruences and minimal presentations Let S = � r 1 , . . . , r k � . a = ( a 1 , . . . , a k ) ∈ N k n = a 1 r 1 + · · · + a k r k � Factorization homomorphism: Monomial map: π : N k − → � r 1 , . . . , r k � ϕ : k [ x 1 , . . . , x k ] − → k [ y ] y r i a �− → a 1 r 1 + · · · + a k r k x i �− → Definition The kernel ker π is the relation ∼ on N k with a ∼ b whenever x a − x b ∈ I S = ker ϕ π ( a ) = π ( b ) ker π is a congruence : an equivalence relation x a − x a = 0 ∈ I S a ∼ a a ∼ b ⇒ b ∼ a a ∼ b and b ∼ c ⇒ a ∼ c that is closed under translation . a ∼ b ⇒ a + c ∼ b + c Christopher O’Neill (UC Davis) Shifting numerical monoids March 21, 2017 9 / 25

  29. Kernel congruences and minimal presentations Let S = � r 1 , . . . , r k � . a = ( a 1 , . . . , a k ) ∈ N k n = a 1 r 1 + · · · + a k r k � Factorization homomorphism: Monomial map: π : N k − → � r 1 , . . . , r k � ϕ : k [ x 1 , . . . , x k ] − → k [ y ] y r i a �− → a 1 r 1 + · · · + a k r k x i �− → Definition The kernel ker π is the relation ∼ on N k with a ∼ b whenever x a − x b ∈ I S = ker ϕ π ( a ) = π ( b ) ker π is a congruence : an equivalence relation x a − x a = 0 ∈ I S a ∼ a x a − x b ∈ I S ⇒ x b − x a ∈ I S a ∼ b ⇒ b ∼ a a ∼ b and b ∼ c ⇒ a ∼ c that is closed under translation . a ∼ b ⇒ a + c ∼ b + c Christopher O’Neill (UC Davis) Shifting numerical monoids March 21, 2017 9 / 25

  30. Kernel congruences and minimal presentations Let S = � r 1 , . . . , r k � . a = ( a 1 , . . . , a k ) ∈ N k n = a 1 r 1 + · · · + a k r k � Factorization homomorphism: Monomial map: π : N k − → � r 1 , . . . , r k � ϕ : k [ x 1 , . . . , x k ] − → k [ y ] y r i a �− → a 1 r 1 + · · · + a k r k x i �− → Definition The kernel ker π is the relation ∼ on N k with a ∼ b whenever x a − x b ∈ I S = ker ϕ π ( a ) = π ( b ) ker π is a congruence : an equivalence relation x a − x a = 0 ∈ I S a ∼ a x a − x b ∈ I S ⇒ x b − x a ∈ I S a ∼ b ⇒ b ∼ a ( x a − x b ) + ( x b − x c ) = x a − x c a ∼ b and b ∼ c ⇒ a ∼ c that is closed under translation . a ∼ b ⇒ a + c ∼ b + c Christopher O’Neill (UC Davis) Shifting numerical monoids March 21, 2017 9 / 25

  31. Kernel congruences and minimal presentations Let S = � r 1 , . . . , r k � . a = ( a 1 , . . . , a k ) ∈ N k n = a 1 r 1 + · · · + a k r k � Factorization homomorphism: Monomial map: π : N k − → � r 1 , . . . , r k � ϕ : k [ x 1 , . . . , x k ] − → k [ y ] y r i a �− → a 1 r 1 + · · · + a k r k x i �− → Definition The kernel ker π is the relation ∼ on N k with a ∼ b whenever x a − x b ∈ I S = ker ϕ π ( a ) = π ( b ) ker π is a congruence : an equivalence relation x a − x a = 0 ∈ I S a ∼ a x a − x b ∈ I S ⇒ x b − x a ∈ I S a ∼ b ⇒ b ∼ a ( x a − x b ) + ( x b − x c ) = x a − x c a ∼ b and b ∼ c ⇒ a ∼ c that is closed under translation . x a − x b ∈ I S ⇒ x c ( x a − x b ) ∈ I S a ∼ b ⇒ a + c ∼ b + c Christopher O’Neill (UC Davis) Shifting numerical monoids March 21, 2017 9 / 25

  32. Kernel congruences and minimal presentations Let S = � r 1 , . . . , r k � . a = ( a 1 , . . . , a k ) ∈ N k n = a 1 r 1 + · · · + a k r k � π : N k − → � r 1 , . . . , r k � �− → a 1 r 1 + · · · + a k r k a Christopher O’Neill (UC Davis) Shifting numerical monoids March 21, 2017 10 / 25

  33. Kernel congruences and minimal presentations Let S = � r 1 , . . . , r k � . a = ( a 1 , . . . , a k ) ∈ N k n = a 1 r 1 + · · · + a k r k � π : N k − → � r 1 , . . . , r k � �− → a 1 r 1 + · · · + a k r k a Definition A minimal presentation ρ of S is a minimal subset ρ ⊂ ker π whose reflexive, symmetric, transitive, and translation closure equals ker π . Christopher O’Neill (UC Davis) Shifting numerical monoids March 21, 2017 10 / 25

  34. Kernel congruences and minimal presentations Let S = � r 1 , . . . , r k � . a = ( a 1 , . . . , a k ) ∈ N k n = a 1 r 1 + · · · + a k r k � π : N k − → � r 1 , . . . , r k � �− → a 1 r 1 + · · · + a k r k a Definition A minimal presentation ρ of S is a minimal subset ρ ⊂ ker π whose reflexive, symmetric, transitive, and translation closure equals ker π . S = � 6 , 9 , 20 � : Christopher O’Neill (UC Davis) Shifting numerical monoids March 21, 2017 10 / 25

  35. Kernel congruences and minimal presentations Let S = � r 1 , . . . , r k � . a = ( a 1 , . . . , a k ) ∈ N k n = a 1 r 1 + · · · + a k r k � π : N k − → � r 1 , . . . , r k � �− → a 1 r 1 + · · · + a k r k a Definition A minimal presentation ρ of S is a minimal subset ρ ⊂ ker π whose reflexive, symmetric, transitive, and translation closure equals ker π . S = � 6 , 9 , 20 � : ρ = { (( 3 , 0 , 0 ) , ( 0 , 2 , 0 )) , (( 4 , 4 , 0 ) , ( 0 , 0 , 3 )) } ⊂ ker π Christopher O’Neill (UC Davis) Shifting numerical monoids March 21, 2017 10 / 25

  36. Kernel congruences and minimal presentations Let S = � r 1 , . . . , r k � . a = ( a 1 , . . . , a k ) ∈ N k n = a 1 r 1 + · · · + a k r k � π : N k − → � r 1 , . . . , r k � �− → a 1 r 1 + · · · + a k r k a Definition A minimal presentation ρ of S is a minimal subset ρ ⊂ ker π whose reflexive, symmetric, transitive, and translation closure equals ker π . S = � 6 , 9 , 20 � : ρ = { (( 3 , 0 , 0 ) , ( 0 , 2 , 0 )) , (( 4 , 4 , 0 ) , ( 0 , 0 , 3 )) } ⊂ ker π π − 1 ( 18 ) : Christopher O’Neill (UC Davis) Shifting numerical monoids March 21, 2017 10 / 25

  37. Kernel congruences and minimal presentations Let S = � r 1 , . . . , r k � . a = ( a 1 , . . . , a k ) ∈ N k n = a 1 r 1 + · · · + a k r k � π : N k − → � r 1 , . . . , r k � �− → a 1 r 1 + · · · + a k r k a Definition A minimal presentation ρ of S is a minimal subset ρ ⊂ ker π whose reflexive, symmetric, transitive, and translation closure equals ker π . S = � 6 , 9 , 20 � : ρ = { (( 3 , 0 , 0 ) , ( 0 , 2 , 0 )) , (( 4 , 4 , 0 ) , ( 0 , 0 , 3 )) } ⊂ ker π π − 1 ( 18 ) : π − 1 ( 60 ) : Christopher O’Neill (UC Davis) Shifting numerical monoids March 21, 2017 10 / 25

  38. Kernel congruences and minimal presentations Let S = � r 1 , . . . , r k � . a = ( a 1 , . . . , a k ) ∈ N k n = a 1 r 1 + · · · + a k r k � π : N k − → � r 1 , . . . , r k � �− → a 1 r 1 + · · · + a k r k a Definition A minimal presentation ρ of S is a minimal subset ρ ⊂ ker π whose reflexive, symmetric, transitive, and translation closure equals ker π . S = � 6 , 9 , 20 � : ρ = { (( 3 , 0 , 0 ) , ( 0 , 2 , 0 )) , (( 4 , 4 , 0 ) , ( 0 , 0 , 3 )) } ⊂ ker π π − 1 ( 18 ) : π − 1 ( 60 ) : (( 7 , 2 , 0 ) , ( 4 , 4 , 0 )) = (( 3 , 0 , 0 ) , ( 0 , 2 , 0 )) + (( 4 , 2 , 0 ) , ( 4 , 2 , 0 )) Christopher O’Neill (UC Davis) Shifting numerical monoids March 21, 2017 10 / 25

  39. Kernel congruences and minimal presentations Let S = � r 1 , . . . , r k � . a = ( a 1 , . . . , a k ) ∈ N k n = a 1 r 1 + · · · + a k r k � π : N k − → � r 1 , . . . , r k � �− → a 1 r 1 + · · · + a k r k a Definition A minimal presentation ρ of S is a minimal subset ρ ⊂ ker π whose reflexive, symmetric, transitive, and translation closure equals ker π . S = � 6 , 9 , 20 � : ρ = { (( 3 , 0 , 0 ) , ( 0 , 2 , 0 )) , (( 4 , 4 , 0 ) , ( 0 , 0 , 3 )) } ⊂ ker π π − 1 ( 18 ) : π − 1 ( 60 ) : (( 7 , 2 , 0 ) , ( 4 , 4 , 0 )) = (( 3 , 0 , 0 ) , ( 0 , 2 , 0 )) + (( 4 , 2 , 0 ) , ( 4 , 2 , 0 )) Cong ( ρ ) = ker π when the graph on π − 1 ( n ) is connected for all n ∈ S . Christopher O’Neill (UC Davis) Shifting numerical monoids March 21, 2017 10 / 25

  40. Kernel congruences and minimal presentations Let S = � r 1 , . . . , r k � . a = ( a 1 , . . . , a k ) ∈ N k n = a 1 r 1 + · · · + a k r k � π : N k − → � r 1 , . . . , r k � �− → a 1 r 1 + · · · + a k r k a Definition A minimal presentation ρ of S is a minimal subset ρ ⊂ ker π whose reflexive, symmetric, transitive, and translation closure equals ker π . S = � 6 , 9 , 20 � : ρ = { (( 3 , 0 , 0 ) , ( 0 , 2 , 0 )) , (( 4 , 4 , 0 ) , ( 0 , 0 , 3 )) } ⊂ ker π π − 1 ( 18 ) : π − 1 ( 60 ) : (( 7 , 2 , 0 ) , ( 4 , 4 , 0 )) = (( 3 , 0 , 0 ) , ( 0 , 2 , 0 )) + (( 4 , 2 , 0 ) , ( 4 , 2 , 0 )) Cong ( ρ ) = ker π when the graph on π − 1 ( n ) is connected for all n ∈ S . I S = � x u − x v : ( u , v ) ∈ ρ � Christopher O’Neill (UC Davis) Shifting numerical monoids March 21, 2017 10 / 25

  41. Kernel congruences and minimal presentations Let S = � r 1 , . . . , r k � . a = ( a 1 , . . . , a k ) ∈ N k n = a 1 r 1 + · · · + a k r k � π : N k − → � r 1 , . . . , r k � �− → a 1 r 1 + · · · + a k r k a Definition A minimal presentation ρ of S is a minimal subset ρ ⊂ ker π whose reflexive, symmetric, transitive, and translation closure equals ker π . S = � 6 , 9 , 20 � : ρ = { (( 3 , 0 , 0 ) , ( 0 , 2 , 0 )) , (( 4 , 4 , 0 ) , ( 0 , 0 , 3 )) } ⊂ ker π π − 1 ( 18 ) : π − 1 ( 60 ) : Christopher O’Neill (UC Davis) Shifting numerical monoids March 21, 2017 11 / 25

  42. Kernel congruences and minimal presentations Let S = � r 1 , . . . , r k � . a = ( a 1 , . . . , a k ) ∈ N k n = a 1 r 1 + · · · + a k r k � π : N k − → � r 1 , . . . , r k � �− → a 1 r 1 + · · · + a k r k a Definition A minimal presentation ρ of S is a minimal subset ρ ⊂ ker π whose reflexive, symmetric, transitive, and translation closure equals ker π . S = � 6 , 9 , 20 � : ρ = { (( 3 , 0 , 0 ) , ( 0 , 2 , 0 )) , (( 4 , 4 , 0 ) , ( 0 , 0 , 3 )) } ⊂ ker π π − 1 ( 18 ) : π − 1 ( 60 ) : All minimal presentations: Christopher O’Neill (UC Davis) Shifting numerical monoids March 21, 2017 11 / 25

  43. Kernel congruences and minimal presentations Let S = � r 1 , . . . , r k � . a = ( a 1 , . . . , a k ) ∈ N k n = a 1 r 1 + · · · + a k r k � π : N k − → � r 1 , . . . , r k � �− → a 1 r 1 + · · · + a k r k a Definition A minimal presentation ρ of S is a minimal subset ρ ⊂ ker π whose reflexive, symmetric, transitive, and translation closure equals ker π . S = � 6 , 9 , 20 � : ρ = { (( 3 , 0 , 0 ) , ( 0 , 2 , 0 )) , (( 4 , 4 , 0 ) , ( 0 , 0 , 3 )) } ⊂ ker π π − 1 ( 18 ) : π − 1 ( 60 ) : All minimal presentations: { (( 3 , 0 , 0 ) , ( 0 , 2 , 0 )) , (( 10 , 7 , 0 ) , ( 0 , 0 , 3 )) } { (( 3 , 0 , 0 ) , ( 0 , 2 , 0 )) , (( 7 , 2 , 0 ) , ( 0 , 0 , 3 )) } { (( 3 , 0 , 0 ) , ( 0 , 2 , 0 )) , (( 4 , 4 , 0 ) , ( 0 , 0 , 3 )) } { (( 3 , 0 , 0 ) , ( 0 , 2 , 0 )) , (( 1 , 6 , 0 ) , ( 0 , 0 , 3 )) } Christopher O’Neill (UC Davis) Shifting numerical monoids March 21, 2017 11 / 25

  44. Kernel congruences and minimal presentations Let S = � r 1 , . . . , r k � . a = ( a 1 , . . . , a k ) ∈ N k n = a 1 r 1 + · · · + a k r k � π : N k − → � r 1 , . . . , r k � �− → a 1 r 1 + · · · + a k r k a Definition A minimal presentation ρ of S is a minimal subset ρ ⊂ ker π whose reflexive, symmetric, transitive, and translation closure equals ker π . S = � 6 , 9 , 20 � : ρ = { (( 3 , 0 , 0 ) , ( 0 , 2 , 0 )) , (( 4 , 4 , 0 ) , ( 0 , 0 , 3 )) } ⊂ ker π π − 1 ( 18 ) : π − 1 ( 60 ) : All minimal presentations: { (( 3 , 0 , 0 ) , ( 0 , 2 , 0 )) , (( 10 , 7 , 0 ) , ( 0 , 0 , 3 )) } { (( 3 , 0 , 0 ) , ( 0 , 2 , 0 )) , (( 7 , 2 , 0 ) , ( 0 , 0 , 3 )) } { (( 3 , 0 , 0 ) , ( 0 , 2 , 0 )) , (( 4 , 4 , 0 ) , ( 0 , 0 , 3 )) } { (( 3 , 0 , 0 ) , ( 0 , 2 , 0 )) , (( 1 , 6 , 0 ) , ( 0 , 0 , 3 )) } Christopher O’Neill (UC Davis) Shifting numerical monoids March 21, 2017 11 / 25

  45. Kernel congruences and minimal presentations Let S = � r 1 , . . . , r k � . a = ( a 1 , . . . , a k ) ∈ N k n = a 1 r 1 + · · · + a k r k � π : N k − → � r 1 , . . . , r k � �− → a 1 r 1 + · · · + a k r k a Definition A minimal presentation ρ of S is a minimal subset ρ ⊂ ker π whose reflexive, symmetric, transitive, and translation closure equals ker π . S = � 6 , 9 , 20 � : ρ = { (( 3 , 0 , 0 ) , ( 0 , 2 , 0 )) , (( 4 , 4 , 0 ) , ( 0 , 0 , 3 )) } ⊂ ker π π − 1 ( 18 ) : π − 1 ( 60 ) : All minimal presentations: { (( 3 , 0 , 0 ) , ( 0 , 2 , 0 )) , (( 10 , 7 , 0 ) , ( 0 , 0 , 3 )) } { (( 3 , 0 , 0 ) , ( 0 , 2 , 0 )) , (( 7 , 2 , 0 ) , ( 0 , 0 , 3 )) } { (( 3 , 0 , 0 ) , ( 0 , 2 , 0 )) , (( 4 , 4 , 0 ) , ( 0 , 0 , 3 )) } { (( 3 , 0 , 0 ) , ( 0 , 2 , 0 )) , (( 1 , 6 , 0 ) , ( 0 , 0 , 3 )) } Christopher O’Neill (UC Davis) Shifting numerical monoids March 21, 2017 11 / 25

  46. Kernel congruences and minimal presentations Let S = � r 1 , . . . , r k � . a = ( a 1 , . . . , a k ) ∈ N k n = a 1 r 1 + · · · + a k r k � π : N k − → � r 1 , . . . , r k � �− → a 1 r 1 + · · · + a k r k a Definition A minimal presentation ρ of S is a minimal subset ρ ⊂ ker π whose reflexive, symmetric, transitive, and translation closure equals ker π . S = � 6 , 9 , 20 � : ρ = { (( 3 , 0 , 0 ) , ( 0 , 2 , 0 )) , (( 4 , 4 , 0 ) , ( 0 , 0 , 3 )) } ⊂ ker π π − 1 ( 18 ) : π − 1 ( 60 ) : All minimal presentations: { (( 3 , 0 , 0 ) , ( 0 , 2 , 0 )) , (( 10 , 7 , 0 ) , ( 0 , 0 , 3 )) } { (( 3 , 0 , 0 ) , ( 0 , 2 , 0 )) , (( 7 , 2 , 0 ) , ( 0 , 0 , 3 )) } { (( 3 , 0 , 0 ) , ( 0 , 2 , 0 )) , (( 4 , 4 , 0 ) , ( 0 , 0 , 3 )) } { (( 3 , 0 , 0 ) , ( 0 , 2 , 0 )) , (( 1 , 6 , 0 ) , ( 0 , 0 , 3 )) } Christopher O’Neill (UC Davis) Shifting numerical monoids March 21, 2017 11 / 25

  47. Kernel congruences and minimal presentations Let S = � r 1 , . . . , r k � . a = ( a 1 , . . . , a k ) ∈ N k n = a 1 r 1 + · · · + a k r k � π : N k − → � r 1 , . . . , r k � �− → a 1 r 1 + · · · + a k r k a Definition A minimal presentation ρ of S is a minimal subset ρ ⊂ ker π whose reflexive, symmetric, transitive, and translation closure equals ker π . S = � 6 , 9 , 20 � : ρ = { (( 3 , 0 , 0 ) , ( 0 , 2 , 0 )) , (( 4 , 4 , 0 ) , ( 0 , 0 , 3 )) } ⊂ ker π π − 1 ( 18 ) : π − 1 ( 60 ) : All minimal presentations: { (( 3 , 0 , 0 ) , ( 0 , 2 , 0 )) , (( 10 , 7 , 0 ) , ( 0 , 0 , 3 )) } { (( 3 , 0 , 0 ) , ( 0 , 2 , 0 )) , (( 7 , 2 , 0 ) , ( 0 , 0 , 3 )) } { (( 3 , 0 , 0 ) , ( 0 , 2 , 0 )) , (( 4 , 4 , 0 ) , ( 0 , 0 , 3 )) } { (( 3 , 0 , 0 ) , ( 0 , 2 , 0 )) , (( 1 , 6 , 0 ) , ( 0 , 0 , 3 )) } Christopher O’Neill (UC Davis) Shifting numerical monoids March 21, 2017 11 / 25

  48. Kernel congruences and minimal presentations Let S = � r 1 , . . . , r k � . a = ( a 1 , . . . , a k ) ∈ N k n = a 1 r 1 + · · · + a k r k � π : N k − → � r 1 , . . . , r k � �− → a 1 r 1 + · · · + a k r k a Definition A minimal presentation ρ of S is a minimal subset ρ ⊂ ker π whose reflexive, symmetric, transitive, and translation closure equals ker π . S = � 6 , 9 , 20 � : ρ = { (( 3 , 0 , 0 ) , ( 0 , 2 , 0 )) , (( 4 , 4 , 0 ) , ( 0 , 0 , 3 )) } ⊂ ker π π − 1 ( 18 ) : π − 1 ( 60 ) : All minimal presentations: { (( 3 , 0 , 0 ) , ( 0 , 2 , 0 )) , (( 10 , 7 , 0 ) , ( 0 , 0 , 3 )) } { (( 3 , 0 , 0 ) , ( 0 , 2 , 0 )) , (( 7 , 2 , 0 ) , ( 0 , 0 , 3 )) } { (( 3 , 0 , 0 ) , ( 0 , 2 , 0 )) , (( 4 , 4 , 0 ) , ( 0 , 0 , 3 )) } { (( 3 , 0 , 0 ) , ( 0 , 2 , 0 )) , (( 1 , 6 , 0 ) , ( 0 , 0 , 3 )) } Christopher O’Neill (UC Davis) Shifting numerical monoids March 21, 2017 11 / 25

  49. Kernel congruences and minimal presentations Let S = � r 1 , . . . , r k � . a = ( a 1 , . . . , a k ) ∈ N k n = a 1 r 1 + · · · + a k r k � π : N k − → � r 1 , . . . , r k � �− → a 1 r 1 + · · · + a k r k a Definition A minimal presentation ρ of S is a minimal subset ρ ⊂ ker π whose reflexive, symmetric, transitive, and translation closure equals ker π . S = � 6 , 9 , 20 � : ρ = { (( 3 , 0 , 0 ) , ( 0 , 2 , 0 )) , (( 4 , 4 , 0 ) , ( 0 , 0 , 3 )) } ⊂ ker π π − 1 ( 18 ) : π − 1 ( 60 ) : All minimal presentations: { (( 3 , 0 , 0 ) , ( 0 , 2 , 0 )) , (( 10 , 7 , 0 ) , ( 0 , 0 , 3 )) } { (( 3 , 0 , 0 ) , ( 0 , 2 , 0 )) , (( 7 , 2 , 0 ) , ( 0 , 0 , 3 )) } { (( 3 , 0 , 0 ) , ( 0 , 2 , 0 )) , (( 4 , 4 , 0 ) , ( 0 , 0 , 3 )) } { (( 3 , 0 , 0 ) , ( 0 , 2 , 0 )) , (( 1 , 6 , 0 ) , ( 0 , 0 , 3 )) } β 0 ( I S ) = { 18 , 60 } Christopher O’Neill (UC Davis) Shifting numerical monoids March 21, 2017 11 / 25

  50. Kernel congruences and minimal presentations π : N k S = � r 1 , . . . , r k � , − → S Christopher O’Neill (UC Davis) Shifting numerical monoids March 21, 2017 12 / 25

  51. Kernel congruences and minimal presentations π : N k S = � r 1 , . . . , r k � , − → S A larger example: S = � 13 , 44 , 106 , 120 � . Christopher O’Neill (UC Davis) Shifting numerical monoids March 21, 2017 12 / 25

  52. Kernel congruences and minimal presentations π : N k S = � r 1 , . . . , r k � , − → S A larger example: S = � 13 , 44 , 106 , 120 � . Minimal presentation ρ has | ρ | = 5 relations. Christopher O’Neill (UC Davis) Shifting numerical monoids March 21, 2017 12 / 25

  53. Kernel congruences and minimal presentations π : N k S = � r 1 , . . . , r k � , − → S A larger example: S = � 13 , 44 , 106 , 120 � . Minimal presentation ρ has | ρ | = 5 relations. Christopher O’Neill (UC Davis) Shifting numerical monoids March 21, 2017 12 / 25

  54. Kernel congruences and minimal presentations π : N k S = � r 1 , . . . , r k � , − → S A larger example: S = � 13 , 44 , 106 , 120 � . Minimal presentation ρ has | ρ | = 5 relations. Christopher O’Neill (UC Davis) Shifting numerical monoids March 21, 2017 12 / 25

  55. Kernel congruences and minimal presentations π : N k S = � r 1 , . . . , r k � , − → S A larger example: S = � 13 , 44 , 106 , 120 � . Minimal presentation ρ has | ρ | = 5 relations. Christopher O’Neill (UC Davis) Shifting numerical monoids March 21, 2017 12 / 25

  56. Kernel congruences and minimal presentations π : N k S = � r 1 , . . . , r k � , − → S A larger example: S = � 13 , 44 , 106 , 120 � . Minimal presentation ρ has | ρ | = 5 relations. Christopher O’Neill (UC Davis) Shifting numerical monoids March 21, 2017 12 / 25

  57. Kernel congruences and minimal presentations π : N k S = � r 1 , . . . , r k � , − → S A larger example: S = � 13 , 44 , 106 , 120 � . Minimal presentation ρ has | ρ | = 5 relations. Christopher O’Neill (UC Davis) Shifting numerical monoids March 21, 2017 12 / 25

  58. Kernel congruences and minimal presentations π : N k S = � r 1 , . . . , r k � , − → S A larger example: S = � 13 , 44 , 106 , 120 � . Minimal presentation ρ has | ρ | = 5 relations. Christopher O’Neill (UC Davis) Shifting numerical monoids March 21, 2017 12 / 25

  59. Kernel congruences and minimal presentations π : N k S = � r 1 , . . . , r k � , − → S A larger example: S = � 13 , 44 , 106 , 120 � . Minimal presentation ρ has | ρ | = 5 relations. Christopher O’Neill (UC Davis) Shifting numerical monoids March 21, 2017 12 / 25

  60. Kernel congruences and minimal presentations π : N k S = � r 1 , . . . , r k � , − → S A larger example: S = � 13 , 44 , 106 , 120 � . Minimal presentation ρ has | ρ | = 5 relations. Christopher O’Neill (UC Davis) Shifting numerical monoids March 21, 2017 12 / 25

  61. Kernel congruences and minimal presentations π : N k S = � r 1 , . . . , r k � , − → S A larger example: S = � 13 , 44 , 106 , 120 � . Minimal presentation ρ has | ρ | = 5 relations. Christopher O’Neill (UC Davis) Shifting numerical monoids March 21, 2017 12 / 25

  62. Kernel congruences and minimal presentations π : N k S = � r 1 , . . . , r k � , − → S A larger example: S = � 13 , 44 , 106 , 120 � . Minimal presentation ρ has | ρ | = 5 relations. Christopher O’Neill (UC Davis) Shifting numerical monoids March 21, 2017 12 / 25

  63. Kernel congruences and minimal presentations π : N k S = � r 1 , . . . , r k � , − → S A larger example: S = � 13 , 44 , 106 , 120 � . Minimal presentation ρ has | ρ | = 5 relations. Christopher O’Neill (UC Davis) Shifting numerical monoids March 21, 2017 12 / 25

  64. Kernel congruences and minimal presentations π : N k S = � r 1 , . . . , r k � , − → S A larger example: S = � 13 , 44 , 106 , 120 � . Minimal presentation ρ has | ρ | = 5 relations. Christopher O’Neill (UC Davis) Shifting numerical monoids March 21, 2017 12 / 25

  65. Intuition: “sufficiently shifted” monoids π n : N k + 1 M n = � n , n + r 1 , . . . , n + r k � , − → M n Christopher O’Neill (UC Davis) Shifting numerical monoids March 21, 2017 13 / 25

  66. Intuition: “sufficiently shifted” monoids π n : N k + 1 M n = � n , n + r 1 , . . . , n + r k � , − → M n Christopher O’Neill (UC Davis) Shifting numerical monoids March 21, 2017 13 / 25

  67. Intuition: “sufficiently shifted” monoids π n : N k + 1 M n = � n , n + r 1 , . . . , n + r k � , − → M n Christopher O’Neill (UC Davis) Shifting numerical monoids March 21, 2017 13 / 25

  68. Intuition: “sufficiently shifted” monoids π n : N k + 1 M n = � n , n + r 1 , . . . , n + r k � , − → M n Christopher O’Neill (UC Davis) Shifting numerical monoids March 21, 2017 13 / 25

  69. Intuition: “sufficiently shifted” monoids π n : N k + 1 M n = � n , n + r 1 , . . . , n + r k � , − → M n Christopher O’Neill (UC Davis) Shifting numerical monoids March 21, 2017 13 / 25

  70. Intuition: “sufficiently shifted” monoids π n : N k + 1 M n = � n , n + r 1 , . . . , n + r k � , − → M n a 0 n + a 1 ( n + r 1 ) + · · · + a k ( n + r k ) = b 0 n + b 1 ( n + r 1 ) + · · · + b k ( n + r k ) Christopher O’Neill (UC Davis) Shifting numerical monoids March 21, 2017 13 / 25

  71. Intuition: “sufficiently shifted” monoids π n : N k + 1 M n = � n , n + r 1 , . . . , n + r k � , − → M n a 0 n + a 1 ( n + r 1 ) + · · · + a k ( n + r k ) = b 0 n + b 1 ( n + r 1 ) + · · · + b k ( n + r k ) Christopher O’Neill (UC Davis) Shifting numerical monoids March 21, 2017 13 / 25

  72. Intuition: “sufficiently shifted” monoids π n : N k + 1 M n = � n , n + r 1 , . . . , n + r k � , − → M n a 0 n + a 1 ( n + r 1 ) + · · · + a k ( n + r k ) = b 0 n + b 1 ( n + r 1 ) + · · · + b k ( n + r k ) | a | n + a 1 r 1 + · · · + a k r k = | b | n + b 1 r 1 + · · · + b k r k Christopher O’Neill (UC Davis) Shifting numerical monoids March 21, 2017 13 / 25

  73. Intuition: “sufficiently shifted” monoids π n : N k + 1 M n = � n , n + r 1 , . . . , n + r k � , − → M n a 0 n + a 1 ( n + r 1 ) + · · · + a k ( n + r k ) = b 0 n + b 1 ( n + r 1 ) + · · · + b k ( n + r k ) | a | n + a 1 r 1 + · · · + a k r k = | b | n + b 1 r 1 + · · · + b k r k 2 types of minimal relations a ∼ b : Christopher O’Neill (UC Davis) Shifting numerical monoids March 21, 2017 13 / 25

  74. Intuition: “sufficiently shifted” monoids π n : N k + 1 M n = � n , n + r 1 , . . . , n + r k � , − → M n a 0 n + a 1 ( n + r 1 ) + · · · + a k ( n + r k ) = b 0 n + b 1 ( n + r 1 ) + · · · + b k ( n + r k ) | a | n + a 1 r 1 + · · · + a k r k = | b | n + b 1 r 1 + · · · + b k r k 2 types of minimal relations a ∼ b : Relations among r 1 , . . . , r k (cheap): | a | = | b | Christopher O’Neill (UC Davis) Shifting numerical monoids March 21, 2017 13 / 25

  75. Intuition: “sufficiently shifted” monoids π n : N k + 1 M n = � n , n + r 1 , . . . , n + r k � , − → M n a 0 n + a 1 ( n + r 1 ) + · · · + a k ( n + r k ) = b 0 n + b 1 ( n + r 1 ) + · · · + b k ( n + r k ) | a | n + a 1 r 1 + · · · + a k r k = | b | n + b 1 r 1 + · · · + b k r k 2 types of minimal relations a ∼ b : Relations among r 1 , . . . , r k (cheap): | a | = | b | Relations that change # copies of n (costly): | a | < | b | Christopher O’Neill (UC Davis) Shifting numerical monoids March 21, 2017 13 / 25

  76. Intuition: “sufficiently shifted” monoids π n : N k + 1 M n = � n , n + r 1 , . . . , n + r k � , − → M n a 0 n + a 1 ( n + r 1 ) + · · · + a k ( n + r k ) = b 0 n + b 1 ( n + r 1 ) + · · · + b k ( n + r k ) | a | n + a 1 r 1 + · · · + a k r k = | b | n + b 1 r 1 + · · · + b k r k 2 types of minimal relations a ∼ b : Relations among r 1 , . . . , r k (cheap): | a | = | b | Relations that change # copies of n (costly): | a | < | b | mostly a k ← − − − − − − → mostly b 0 Christopher O’Neill (UC Davis) Shifting numerical monoids March 21, 2017 13 / 25

  77. Intuition: “sufficiently shifted” monoids π n : N k + 1 M n = � n , n + r 1 , . . . , n + r k � , − → M n a 0 n + a 1 ( n + r 1 ) + · · · + a k ( n + r k ) = b 0 n + b 1 ( n + r 1 ) + · · · + b k ( n + r k ) | a | n + a 1 r 1 + · · · + a k r k = | b | n + b 1 r 1 + · · · + b k r k 2 types of minimal relations a ∼ b : Relations among r 1 , . . . , r k (cheap): | a | = | b | Relations that change # copies of n (costly): | a | < | b | mostly a k ← − − − − − − → mostly b 0 In M n = � n , n + 6 , n + 9 , n + 20 � with n = 450: Christopher O’Neill (UC Davis) Shifting numerical monoids March 21, 2017 13 / 25

  78. Intuition: “sufficiently shifted” monoids π n : N k + 1 M n = � n , n + r 1 , . . . , n + r k � , − → M n a 0 n + a 1 ( n + r 1 ) + · · · + a k ( n + r k ) = b 0 n + b 1 ( n + r 1 ) + · · · + b k ( n + r k ) | a | n + a 1 r 1 + · · · + a k r k = | b | n + b 1 r 1 + · · · + b k r k 2 types of minimal relations a ∼ b : Relations among r 1 , . . . , r k (cheap): | a | = | b | Relations that change # copies of n (costly): | a | < | b | mostly a k ← − − − − − − → mostly b 0 In M n = � n , n + 6 , n + 9 , n + 20 � with n = 450: 3 ( n + 6 ) = n + 2 ( n + 9 ) is cheap 4 ( n + 9 ) + 21 ( n + 20 ) = 25 n + ( n + 6 ) is costly Christopher O’Neill (UC Davis) Shifting numerical monoids March 21, 2017 13 / 25

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