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Minimal presentations of shifted 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 * =

  1. 250 0 15 10 5 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 � . 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) Shifted numerical monoids Sep 26, 2016 4 / 23

  2. 250 0 15 10 5 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 � . 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) Shifted numerical monoids Sep 26, 2016 4 / 23

  3. 200 250 15 10 5 0 50 100 150 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) Shifted numerical monoids Sep 26, 2016 4 / 23

  4. 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) Shifted numerical monoids Sep 26, 2016 5 / 23

  5. 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 . Christopher O’Neill (UC Davis) Shifted numerical monoids Sep 26, 2016 5 / 23

  6. 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 . Christopher O’Neill (UC Davis) Shifted numerical monoids Sep 26, 2016 5 / 23

  7. 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) Shifted numerical monoids Sep 26, 2016 5 / 23

  8. 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) Shifted numerical monoids Sep 26, 2016 5 / 23

  9. 0 2000 4000 50 100 150 200 250 3000 1000 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) Shifted numerical monoids Sep 26, 2016 6 / 23

  10. 0 1000 4000 3000 50 100 150 200 250 2000 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) Shifted numerical monoids Sep 26, 2016 6 / 23

  11. 250 0 4000 3000 2000 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. Christopher O’Neill (UC Davis) Shifted numerical monoids Sep 26, 2016 6 / 23

  12. 200 250 4000 3000 2000 1000 0 50 100 150 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) Shifted numerical monoids Sep 26, 2016 6 / 23

  13. 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) Shifted numerical monoids Sep 26, 2016 7 / 23

  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: Christopher O’Neill (UC Davis) Shifted numerical monoids Sep 26, 2016 7 / 23

  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. Christopher O’Neill (UC Davis) Shifted numerical monoids Sep 26, 2016 7 / 23

  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. Christopher O’Neill (UC Davis) Shifted numerical monoids Sep 26, 2016 7 / 23

  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. Christopher O’Neill (UC Davis) Shifted numerical monoids Sep 26, 2016 7 / 23

  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: Christopher O’Neill (UC Davis) Shifted numerical monoids Sep 26, 2016 7 / 23

  19. 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) Shifted numerical monoids Sep 26, 2016 7 / 23

  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 � Christopher O’Neill (UC Davis) Shifted numerical monoids Sep 26, 2016 8 / 23

  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 Christopher O’Neill (UC Davis) Shifted numerical monoids Sep 26, 2016 8 / 23

  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 ) Christopher O’Neill (UC Davis) Shifted numerical monoids Sep 26, 2016 8 / 23

  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 Christopher O’Neill (UC Davis) Shifted numerical monoids Sep 26, 2016 8 / 23

  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) Shifted numerical monoids Sep 26, 2016 8 / 23

  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: 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) Shifted numerical monoids Sep 26, 2016 9 / 23

  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 � 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) Shifted numerical monoids Sep 26, 2016 9 / 23

  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 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) Shifted numerical monoids Sep 26, 2016 9 / 23

  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 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) Shifted numerical monoids Sep 26, 2016 9 / 23

  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 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) Shifted numerical monoids Sep 26, 2016 9 / 23

  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 a ∼ b and b ∼ c ⇒ a ∼ c that is closed under translation . a ∼ b ⇒ a + c ∼ b + c Christopher O’Neill (UC Davis) Shifted numerical monoids Sep 26, 2016 9 / 23

  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 . a ∼ b ⇒ a + c ∼ b + c Christopher O’Neill (UC Davis) Shifted numerical monoids Sep 26, 2016 9 / 23

  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 � 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) Shifted numerical monoids Sep 26, 2016 9 / 23

  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 Christopher O’Neill (UC Davis) Shifted numerical monoids Sep 26, 2016 10 / 23

  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 . Christopher O’Neill (UC Davis) Shifted numerical monoids Sep 26, 2016 10 / 23

  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 � : Christopher O’Neill (UC Davis) Shifted numerical monoids Sep 26, 2016 10 / 23

  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 Christopher O’Neill (UC Davis) Shifted numerical monoids Sep 26, 2016 10 / 23

  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): Christopher O’Neill (UC Davis) Shifted numerical monoids Sep 26, 2016 10 / 23

  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): Christopher O’Neill (UC Davis) Shifted numerical monoids Sep 26, 2016 10 / 23

  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)) Christopher O’Neill (UC Davis) Shifted numerical monoids Sep 26, 2016 10 / 23

  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 . Christopher O’Neill (UC Davis) Shifted numerical monoids Sep 26, 2016 10 / 23

  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): ((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) Shifted numerical monoids Sep 26, 2016 10 / 23

  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): Christopher O’Neill (UC Davis) Shifted numerical monoids Sep 26, 2016 11 / 23

  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: Christopher O’Neill (UC Davis) Shifted numerical monoids Sep 26, 2016 11 / 23

  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) Shifted numerical monoids Sep 26, 2016 11 / 23

  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) Shifted numerical monoids Sep 26, 2016 11 / 23

  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) Shifted numerical monoids Sep 26, 2016 11 / 23

  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) Shifted numerical monoids Sep 26, 2016 11 / 23

  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) Shifted numerical monoids Sep 26, 2016 11 / 23

  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)) } Christopher O’Neill (UC Davis) Shifted numerical monoids Sep 26, 2016 11 / 23

  50. 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) Shifted numerical monoids Sep 26, 2016 11 / 23

  51. 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) Shifted numerical monoids Sep 26, 2016 12 / 23

  52. 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) Shifted numerical monoids Sep 26, 2016 12 / 23

  53. 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) Shifted numerical monoids Sep 26, 2016 12 / 23

  54. 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) Shifted numerical monoids Sep 26, 2016 12 / 23

  55. 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) Shifted numerical monoids Sep 26, 2016 12 / 23

  56. Intuition: “sufficiently shifted” monoids π n : N k +1 M n = � n , n + r 1 , . . . , n + r k � , − → M n n + ( n + r 1 ) + · · · + ( n + r k ) Christopher O’Neill (UC Davis) Shifted numerical monoids Sep 26, 2016 12 / 23

  57. Intuition: “sufficiently shifted” monoids π n : N k +1 M n = � n , n + r 1 , . . . , n + r k � , − → M n n + ( n + r 1 ) + · · · + ( n + r k ) Christopher O’Neill (UC Davis) Shifted numerical monoids Sep 26, 2016 12 / 23

  58. Intuition: “sufficiently shifted” monoids π n : N k +1 M n = � n , n + r 1 , . . . , n + r k � , − → M n n + ( n + r 1 ) + · · · + ( n + r k ) 2 types of minimal relations a ∼ b : Christopher O’Neill (UC Davis) Shifted numerical monoids Sep 26, 2016 12 / 23

  59. Intuition: “sufficiently shifted” monoids π n : N k +1 M n = � n , n + r 1 , . . . , n + r k � , − → M n n + ( n + r 1 ) + · · · + ( n + r k ) 2 types of minimal relations a ∼ b : Relations among r 1 , . . . , r k (cheap) Christopher O’Neill (UC Davis) Shifted numerical monoids Sep 26, 2016 12 / 23

  60. Intuition: “sufficiently shifted” monoids π n : N k +1 M n = � n , n + r 1 , . . . , n + r k � , − → M n n + ( n + r 1 ) + · · · + ( n + r k ) 2 types of minimal relations a ∼ b : Relations among r 1 , . . . , r k (cheap) | a | = | b | Christopher O’Neill (UC Davis) Shifted numerical monoids Sep 26, 2016 12 / 23

  61. Intuition: “sufficiently shifted” monoids π n : N k +1 M n = � n , n + r 1 , . . . , n + r k � , − → M n n + ( n + r 1 ) + · · · + ( n + 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) Christopher O’Neill (UC Davis) Shifted numerical monoids Sep 26, 2016 12 / 23

  62. Intuition: “sufficiently shifted” monoids π n : N k +1 M n = � n , n + r 1 , . . . , n + r k � , − → M n n + ( n + r 1 ) + · · · + ( n + 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) mostly a k ← − − − − − − → mostly b 0 Christopher O’Neill (UC Davis) Shifted numerical monoids Sep 26, 2016 12 / 23

  63. The shifting map π n : N k +1 M n = � n , n + r 1 , . . . , n + r k � , − → M n Christopher O’Neill (UC Davis) Shifted numerical monoids Sep 26, 2016 13 / 23

  64. The shifting map π n : N k +1 M n = � n , n + r 1 , . . . , n + r k � , − → M n Shifting map Φ n : ker π n − → ker π n + r k given by Christopher O’Neill (UC Davis) Shifted numerical monoids Sep 26, 2016 13 / 23

  65. The shifting map π n : N k +1 M n = � n , n + r 1 , . . . , n + r k � , − → M n Shifting map Φ n : ker π n − → ker π n + r k given by  ( a , a ′ ) | a | = | a ′ |  ( a + ℓ e k , a ′ + ℓ e 0 ) ( a , a ′ ) �− → | a | < | a ′ | ( a + ℓ e 0 , a ′ + ℓ e k )  | a | > | a ′ | � � where ℓ = � | a | − | a ′ | � . Christopher O’Neill (UC Davis) Shifted numerical monoids Sep 26, 2016 13 / 23

  66. The shifting map π n : N k +1 M n = � n , n + r 1 , . . . , n + r k � , − → M n Shifting map Φ n : ker π n − → ker π n + r k given by  ( a , a ′ ) | a | = | a ′ |  ( a + ℓ e k , a ′ + ℓ e 0 ) ( a , a ′ ) �− → | a | < | a ′ | ( a + ℓ e 0 , a ′ + ℓ e k )  | a | > | a ′ | � � where ℓ = � | a | − | a ′ | � . DON’T PANIC! Christopher O’Neill (UC Davis) Shifted numerical monoids Sep 26, 2016 13 / 23

  67. The shifting map π n : N k +1 M n = � n , n + r 1 , . . . , n + r k � , − → M n Shifting map Φ n : ker π n − → ker π n + r k given by  ( a , a ′ ) | a | = | a ′ |  ( a + ℓ e k , a ′ + ℓ e 0 ) ( a , a ′ ) �− → | a | < | a ′ | ( a + ℓ e 0 , a ′ + ℓ e k )  | a | > | a ′ | � � where ℓ = � | a | − | a ′ | � . Christopher O’Neill (UC Davis) Shifted numerical monoids Sep 26, 2016 14 / 23

  68. The shifting map π n : N k +1 M n = � n , n + r 1 , . . . , n + r k � , − → M n Shifting map Φ n : ker π n − → ker π n + r k given by  ( a , a ′ ) | a | = | a ′ |  ( a + ℓ e k , a ′ + ℓ e 0 ) ( a , a ′ ) �− → | a | < | a ′ | ( a + ℓ e 0 , a ′ + ℓ e k )  | a | > | a ′ | � � where ℓ = � | a | − | a ′ | � . Sneak peek for M n = � n , n + 6 , n + 9 , n + 20 � and n ≫ 0: Christopher O’Neill (UC Davis) Shifted numerical monoids Sep 26, 2016 14 / 23

  69. The shifting map π n : N k +1 M n = � n , n + r 1 , . . . , n + r k � , − → M n Shifting map Φ n : ker π n − → ker π n + r k given by  ( a , a ′ ) | a | = | a ′ |  ( a + ℓ e k , a ′ + ℓ e 0 ) ( a , a ′ ) �− → | a | < | a ′ | ( a + ℓ e 0 , a ′ + ℓ e k )  | a | > | a ′ | � � where ℓ = � | a | − | a ′ | � . Sneak peek for M n = � n , n + 6 , n + 9 , n + 20 � and n ≫ 0: M 450 : � (( 0 , 0 , 8 , 0) , (3 , 2 , 0 , 3)) , (( 0 , 1 , 6 , 0) , (4 , 0 , 0 , 3)) , (( 0 , 3 , 0 , 0) , (1 , 0 , 2 , 0)) , � ((20 , 5 , 0 , 0) , (0 , 0 , 0 , 24)) , ((25 , 1 , 0 , 0) , (0 , 0 , 4 , 21)) , ((26 , 0 , 0 , 0) , (0 , 2 , 2 , 21)) Christopher O’Neill (UC Davis) Shifted numerical monoids Sep 26, 2016 14 / 23

  70. The shifting map π n : N k +1 M n = � n , n + r 1 , . . . , n + r k � , − → M n Shifting map Φ n : ker π n − → ker π n + r k given by  ( a , a ′ ) | a | = | a ′ |  ( a + ℓ e k , a ′ + ℓ e 0 ) ( a , a ′ ) �− → | a | < | a ′ | ( a + ℓ e 0 , a ′ + ℓ e k )  | a | > | a ′ | � � where ℓ = � | a | − | a ′ | � . Sneak peek for M n = � n , n + 6 , n + 9 , n + 20 � and n ≫ 0: M 450 : � (( 0 , 0 , 8 , 0) , (3 , 2 , 0 , 3)) , (( 0 , 1 , 6 , 0) , (4 , 0 , 0 , 3)) , (( 0 , 3 , 0 , 0) , (1 , 0 , 2 , 0)) , � ((20 , 5 , 0 , 0) , (0 , 0 , 0 , 24)) , ((25 , 1 , 0 , 0) , (0 , 0 , 4 , 21)) , ((26 , 0 , 0 , 0) , (0 , 2 , 2 , 21)) M 470 : � (( 0 , 0 , 8 , 0) , (3 , 2 , 0 , 3)) , (( 0 , 1 , 6 , 0) , (4 , 0 , 0 , 3)) , (( 0 , 3 , 0 , 0) , (1 , 0 , 2 , 0)) , � ((21 , 5 , 0 , 0) , (0 , 0 , 0 , 25)) , ((26 , 1 , 0 , 0) , (0 , 0 , 4 , 22)) , ((27 , 0 , 0 , 0) , (0 , 2 , 2 , 22)) Christopher O’Neill (UC Davis) Shifted numerical monoids Sep 26, 2016 14 / 23

  71. The shifting map π n : N k +1 M n = � n , n + r 1 , . . . , n + r k � , − → M n Shifting map Φ n : ker π n − → ker π n + r k given by  ( a , a ′ ) | a | = | a ′ |  ( a + ℓ e k , a ′ + ℓ e 0 ) ( a , a ′ ) �− → | a | < | a ′ | ( a + ℓ e 0 , a ′ + ℓ e k )  | a | > | a ′ | � � where ℓ = � | a | − | a ′ | � . Sneak peek for M n = � n , n + 6 , n + 9 , n + 20 � and n ≫ 0: M 450 : � (( 0 , 0 , 8 , 0) , (3 , 2 , 0 , 3)) , (( 0 , 1 , 6 , 0) , (4 , 0 , 0 , 3)) , (( 0 , 3 , 0 , 0) , (1 , 0 , 2 , 0)) , � ((20 , 5 , 0 , 0) , (0 , 0 , 0 , 24)) , ((25 , 1 , 0 , 0) , (0 , 0 , 4 , 21)) , ((26 , 0 , 0 , 0) , (0 , 2 , 2 , 21)) M 470 : � (( 0 , 0 , 8 , 0) , (3 , 2 , 0 , 3)) , (( 0 , 1 , 6 , 0) , (4 , 0 , 0 , 3)) , (( 0 , 3 , 0 , 0) , (1 , 0 , 2 , 0)) , � ((21 , 5 , 0 , 0) , (0 , 0 , 0 , 25)) , ((26 , 1 , 0 , 0) , (0 , 0 , 4 , 22)) , ((27 , 0 , 0 , 0) , (0 , 2 , 2 , 22)) M 490 : � (( 0 , 0 , 8 , 0) , (3 , 2 , 0 , 3)) , (( 0 , 1 , 6 , 0) , (4 , 0 , 0 , 3)) , (( 0 , 3 , 0 , 0) , (1 , 0 , 2 , 0)) , � ((22 , 5 , 0 , 0) , (0 , 0 , 0 , 26)) , ((27 , 1 , 0 , 0) , (0 , 0 , 4 , 23)) , ((28 , 0 , 0 , 0) , (0 , 2 , 2 , 23)) Christopher O’Neill (UC Davis) Shifted numerical monoids Sep 26, 2016 14 / 23

  72. The shifting map π n : N k +1 M n = � n , n + r 1 , . . . , n + r k � , − → M n Shifting map Φ n : ker π n − → ker π n + r k given by  ( a , a ′ ) | a | = | a ′ |  ( a + ℓ e k , a ′ + ℓ e 0 ) ( a , a ′ ) �− → | a | < | a ′ | ( a + ℓ e 0 , a ′ + ℓ e k )  | a | > | a ′ | � � where ℓ = � | a | − | a ′ | � . Christopher O’Neill (UC Davis) Shifted numerical monoids Sep 26, 2016 15 / 23

  73. The shifting map π n : N k +1 M n = � n , n + r 1 , . . . , n + r k � , − → M n Shifting map Φ n : ker π n − → ker π n + r k given by  ( a , a ′ ) | a | = | a ′ |  ( a + ℓ e k , a ′ + ℓ e 0 ) ( a , a ′ ) �− → | a | < | a ′ | ( a + ℓ e 0 , a ′ + ℓ e k )  | a | > | a ′ | � � where ℓ = � | a | − | a ′ | � . Φ n is well-defined. Christopher O’Neill (UC Davis) Shifted numerical monoids Sep 26, 2016 15 / 23

  74. The shifting map π n : N k +1 M n = � n , n + r 1 , . . . , n + r k � , − → M n Shifting map Φ n : ker π n − → ker π n + r k given by  ( a , a ′ ) | a | = | a ′ |  ( a + ℓ e k , a ′ + ℓ e 0 ) ( a , a ′ ) �− → | a | < | a ′ | ( a + ℓ e 0 , a ′ + ℓ e k )  | a | > | a ′ | � � where ℓ = � | a | − | a ′ | � . Φ n is well-defined. π n ( a ) = a 0 n + � k i =1 a i ( n + r i ) = | a | n + � k i =1 a i r i = | a | n + | a | r k + � k π n + r k ( a ) = i =1 a i r i Christopher O’Neill (UC Davis) Shifted numerical monoids Sep 26, 2016 15 / 23

  75. The shifting map π n : N k +1 M n = � n , n + r 1 , . . . , n + r k � , − → M n Shifting map Φ n : ker π n − → ker π n + r k given by  ( a , a ′ ) | a | = | a ′ |  ( a + ℓ e k , a ′ + ℓ e 0 ) ( a , a ′ ) �− → | a | < | a ′ | ( a + ℓ e 0 , a ′ + ℓ e k )  | a | > | a ′ | � � where ℓ = � | a | − | a ′ | � . Φ n is well-defined. π n ( a ) = a 0 n + � k i =1 a i ( n + r i ) = | a | n + � k i =1 a i r i = | a | n + | a | r k + � k π n + r k ( a ) = i =1 a i r i Φ n preserves reflexive and symmetric closure. Christopher O’Neill (UC Davis) Shifted numerical monoids Sep 26, 2016 15 / 23

  76. The shifting map π n : N k +1 M n = � n , n + r 1 , . . . , n + r k � , − → M n Shifting map Φ n : ker π n − → ker π n + r k given by  ( a , a ′ ) | a | = | a ′ |  ( a + ℓ e k , a ′ + ℓ e 0 ) ( a , a ′ ) �− → | a | < | a ′ | ( a + ℓ e 0 , a ′ + ℓ e k )  | a | > | a ′ | � � where ℓ = � | a | − | a ′ | � . Φ n is well-defined. π n ( a ) = a 0 n + � k i =1 a i ( n + r i ) = | a | n + � k i =1 a i r i = | a | n + | a | r k + � k π n + r k ( a ) = i =1 a i r i Φ n preserves reflexive and symmetric closure. Φ n preserves translation closure. Christopher O’Neill (UC Davis) Shifted numerical monoids Sep 26, 2016 15 / 23

  77. The shifting map π n : N k +1 M n = � n , n + r 1 , . . . , n + r k � , − → M n Shifting map Φ n : ker π n − → ker π n + r k given by  ( a , a ′ ) | a | = | a ′ |  ( a + ℓ e k , a ′ + ℓ e 0 ) ( a , a ′ ) �− → | a | < | a ′ | ( a + ℓ e 0 , a ′ + ℓ e k )  | a | > | a ′ | � � where ℓ = � | a | − | a ′ | � . Φ n is well-defined. π n ( a ) = a 0 n + � k i =1 a i ( n + r i ) = | a | n + � k i =1 a i r i = | a | n + | a | r k + � k π n + r k ( a ) = i =1 a i r i Φ n preserves reflexive and symmetric closure. Φ n preserves translation closure. Φ n (( a , a ′ ) + ( b , b )) = Φ n ( a , a ′ ) + ( b , b ) Christopher O’Neill (UC Davis) Shifted numerical monoids Sep 26, 2016 15 / 23

  78. The shifting map π n : N k +1 M n = � n , n + r 1 , . . . , n + r k � , − → M n Shifting map Φ n : ker π n − → ker π n + r k given by  ( a , a ′ ) | a | = | a ′ |  ( a + ℓ e k , a ′ + ℓ e 0 ) ( a , a ′ ) �− → | a | < | a ′ | ( a + ℓ e 0 , a ′ + ℓ e k )  | a | > | a ′ | � � where ℓ = � | a | − | a ′ | � . Φ n is well-defined. π n ( a ) = a 0 n + � k i =1 a i ( n + r i ) = | a | n + � k i =1 a i r i = | a | n + | a | r k + � k π n + r k ( a ) = i =1 a i r i Φ n preserves reflexive and symmetric closure. Φ n preserves translation closure. Φ n (( a , a ′ ) + ( b , b )) = Φ n ( a , a ′ ) + ( b , b ) Only missing link: transitivity. Christopher O’Neill (UC Davis) Shifted numerical monoids Sep 26, 2016 15 / 23

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