shifting numerical monoids
play

Shifting numerical monoids Christopher ONeill University of - PowerPoint PPT Presentation

Mar 10, 2023 •158 likes •1.93k views

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

Recommend

Catenary degree in numerical. monoids (An application to numerical monoids generated by

Catenary degree in numerical. monoids (An application to numerical monoids generated by

Catenary degree in numerical. monoids (An application to numerical monoids generated by arithmetic sequences) D. Llena Carrasco Area de Geometr a y Topolog a Universidad de Almer a Porto-March 2008 Catenary degree Credits

517 views • 49 slides
The Catenary Degree of Numerical Monoids and Krull Monoids Alfred Geroldinger Institute of

The Catenary Degree of Numerical Monoids and Krull Monoids Alfred Geroldinger Institute of

The Catenary Degree of Numerical Monoids and Krull Monoids Alfred Geroldinger Institute of Mathematics and Scientific Computing University of Graz February 2010 Iberian Meeting on Numerical Semigroups Facultad de Ciencias, Universidad de

661 views • 25 slides
Cyclic rational semirings A comparison between the factorization invariants of numerical monoids

Cyclic rational semirings A comparison between the factorization invariants of numerical monoids

Cyclic rational semirings A comparison between the factorization invariants of numerical monoids and Puiseux monoids Marly Gotti March 22, 2019 AMS Sectional Meeting, University of Hawaii at Manoa Table of contents 1. General comparison

630 views • 60 slides
Minimal presentations of shifted numerical monoids Christopher ONeill Texas A&amp;M University

Minimal presentations of shifted numerical monoids Christopher ONeill Texas A&amp;M University

Minimal presentations of shifted numerical monoids Christopher ONeill Texas A&amp;M University coneill@math.tamu.edu Joint with Rebecca Conaway*, Felix Gotti, Jesse Horton*, Roberto Pelayo, Mesa Williams*, and Brian Wissman * =

964 views • 83 slides
Minimal presentations of shifted numerical monoids Christopher ONeill University of California

Minimal presentations of shifted numerical monoids Christopher ONeill University of California

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 * =

1.82k views • 159 slides
Computing the delta set and -primality in numerical monoids Christopher ONeill Texas

Computing the delta set and -primality in numerical monoids Christopher ONeill Texas

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

1.19k views • 114 slides
Catenary degrees of elements in numerical monoids Christopher ONeill Texas A&amp;M University

Catenary degrees of elements in numerical monoids Christopher ONeill Texas A&amp;M University

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

1.16k views • 114 slides
Puiseux Monoids and Their Atomic Structure Felix Gotti felixgotti@berkeley.edu UC Berkeley

Puiseux Monoids and Their Atomic Structure Felix Gotti felixgotti@berkeley.edu UC Berkeley

Puiseux Monoids and Their Atomic Structure Felix Gotti felixgotti@berkeley.edu UC Berkeley International Meeting on Numerical Semigroups July 6, 2016 Felix Gotti felixgotti@berkeley.edu Puiseux Monoids and Their Atomic Structure Outline

998 views • 77 slides
Numerical semigroups in Sage Christopher ONeill University of California Davis

Numerical semigroups in Sage Christopher ONeill University of California Davis

Numerical semigroups in Sage Christopher ONeill University of California Davis coneill@math.ucdavis.edu Oct 26, 2016 Christopher ONeill (UC Davis) Numerical semigroups Oct 26, 2016 1 / 9 Numerical monoids Definition A numerical

575 views • 42 slides
Factorization in complement-finite ideals of free monoids Nicholas R. Baeth (Joint work with

Factorization in complement-finite ideals of free monoids Nicholas R. Baeth (Joint work with

Factorization in complement-finite ideals of free monoids Nicholas R. Baeth (Joint work with Matthew Enlow) March 23, 2019 Franklin &amp; Marshall College Multiplication in Numerical Semigroups Let S = n 1 , . . . , n t = { a 1 n 1 +

525 views • 21 slides
Conductor ideals of affine monoids and K -theory Joseph Gubeladze San Francisco State University

Conductor ideals of affine monoids and K -theory Joseph Gubeladze San Francisco State University

Conductor ideals of affine monoids and K -theory Joseph Gubeladze San Francisco State University AMS Special Session: Combinatorial Ideals and Applications Fargo, 2016 Outline Frobenius number of a numerical semigroup Affine monoid,

644 views • 47 slides
Pattern avoidance Definitions in rook monoids Rook Monoids Avoidance 1d Avoidance All 0/No 0

Pattern avoidance Definitions in rook monoids Rook Monoids Avoidance 1d Avoidance All 0/No 0

Pattern avoidance in rook monoids Lara Pudwell Pattern avoidance Definitions in rook monoids Rook Monoids Avoidance 1d Avoidance All 0/No 0 Dan Daly (Southeast Missouri State University) patterns Other patterns Lara Pudwell (Valparaiso

890 views • 32 slides
Circuit Complexity of Regular Languages Michal Koucky Presented by, Sunil K. S April 13, 2012

Circuit Complexity of Regular Languages Michal Koucky Presented by, Sunil K. S April 13, 2012

Circuit Complexity of Regular Languages Michal Koucky Presented by, Sunil K. S April 13, 2012 1 / 28 Contents Theme of Presentation Algebraic Preliminaries: Monoids Operations on Monoids Monoids as Recognizers Monoids and Automata

2.56k views • 35 slides
On Numerical Semigroups Maria Bras-Amors Universitat Rovira i Virgili, Catalonia Spring

On Numerical Semigroups Maria Bras-Amors Universitat Rovira i Virgili, Catalonia Spring

On Numerical Semigroups Maria Bras-Amors Universitat Rovira i Virgili, Catalonia Spring Central and Western Joint Sectional Meeting of the AMS Special Session on Factorization and Arithmetic Properties of Integral Domains and Monoids March

1.2k views • 99 slides
Topological finiteness properties of monoids Robert D. Gray 1 (joint work with B. Steinberg (City

Topological finiteness properties of monoids Robert D. Gray 1 (joint work with B. Steinberg (City

Topological finiteness properties of monoids Robert D. Gray 1 (joint work with B. Steinberg (City College of New York)) SLADIM+ seminar Novi Sad, February 2018 1 Research supported by the EPSRC grant EP/N033353/1 &quot;Special inverse monoids:

588 views • 37 slides
Lecture 3 Interacting Hopf monoids and graphical linear algebra Plan relational intuitions

Lecture 3 Interacting Hopf monoids and graphical linear algebra Plan relational intuitions

Lecture 3 Interacting Hopf monoids and graphical linear algebra Plan relational intuitions Frobenius monoids the equations of interacting Hopf monoids linear relations rational numbers, diagrammatically Relational intuitions

516 views • 41 slides
Varieties of De Morgan Monoids II: Covers of Atoms T. Moraschini 1 , J.G. Raftery 2 , and J.J.

Varieties of De Morgan Monoids II: Covers of Atoms T. Moraschini 1 , J.G. Raftery 2 , and J.J.

Varieties of De Morgan Monoids II: Covers of Atoms T. Moraschini 1 , J.G. Raftery 2 , and J.J. Wannenburg 2 1 Academy of Sciences of the Czech Republic, Czech Republic 2 University of Pretoria, South Africa TACL, June 2017 De Morgan monoids A De

876 views • 39 slides
Patterns of ideals of numerical semigroups Klara Stokes AMS Sectional Meeting Special Session

Patterns of ideals of numerical semigroups Klara Stokes AMS Sectional Meeting Special Session

Patterns of ideals of numerical semigroups Klara Stokes AMS Sectional Meeting Special Session on Factorization and Arithmetic Properties of Integral Domains and Monoids March 24, 2019 Table of Contents Introduction 1 Patterns of ideals of

1.01k views • 62 slides
Hopf monoids in duoidal categories Gabriella B ohm Wigner Research Centre for Physics

Hopf monoids in duoidal categories Gabriella B ohm Wigner Research Centre for Physics

Hopf monoids in duoidal categories Gabriella B ohm Wigner Research Centre for Physics Category Theory 2015, Aveiro 16th of June Plan 1. What distinguishes groups among monoids ? 2. A universal approach: bimonoids in duoidal categories

782 views • 47 slides
Varieties of De Morgan Monoids T. Moraschini 1 , J.G. Raftery 2 , and J.J. Wannenburg 2 1 Academy

Varieties of De Morgan Monoids T. Moraschini 1 , J.G. Raftery 2 , and J.J. Wannenburg 2 1 Academy

Varieties of De Morgan Monoids T. Moraschini 1 , J.G. Raftery 2 , and J.J. Wannenburg 2 1 Academy of Sciences of the Czech Republic, Czech Republic 2 University of Pretoria, South Africa SAMSA, November 2016 De Morgan monoids A De Morgan monoid

1.1k views • 72 slides
Numerical Differentiation &amp; Integration Numerical Differentiation I Numerical Analysis (9th

Numerical Differentiation &amp; Integration Numerical Differentiation I Numerical Analysis (9th

Numerical Differentiation &amp; Integration Numerical Differentiation I Numerical Analysis (9th Edition) R L Burden &amp; J D Faires Beamer Presentation Slides prepared by John Carroll Dublin City University c 2011 Brooks/Cole, Cengage

774 views • 64 slides
Numerical Differentiation &amp; Integration Numerical Differentiation II Numerical Analysis (9th

Numerical Differentiation &amp; Integration Numerical Differentiation II Numerical Analysis (9th

Numerical Differentiation &amp; Integration Numerical Differentiation II Numerical Analysis (9th Edition) R L Burden &amp; J D Faires Beamer Presentation Slides prepared by John Carroll Dublin City University c 2011 Brooks/Cole, Cengage

825 views • 43 slides
Numerical Differentiation &amp; Integration Elements of Numerical Integration I Numerical

Numerical Differentiation &amp; Integration Elements of Numerical Integration I Numerical

Numerical Differentiation &amp; Integration Elements of Numerical Integration I Numerical Analysis (9th Edition) R L Burden &amp; J D Faires Beamer Presentation Slides prepared by John Carroll Dublin City University c 2011 Brooks/Cole,

1.24k views • 81 slides
Families of Numerical Semigroups Kunz Coordinates and Semigroup Trees Nathan Kaplan University

Families of Numerical Semigroups Kunz Coordinates and Semigroup Trees Nathan Kaplan University

Families of Numerical Semigroups Kunz Coordinates and Semigroup Trees Nathan Kaplan University of California, Irvine AMS 2019: Factorization and Arithmetic Properties of Integral Domains and Monoids March 22, 2019 Kaplan (UCI) Families of

502 views • 19 slides

More recommend