Numerical semigroups in Sage Christopher ONeill University of - PowerPoint PPT Presentation

Numerical semigroups in Sage Christopher O’Neill University of California Davis coneill@math.ucdavis.edu Oct 26, 2016 Christopher O’Neill (UC Davis) Numerical semigroups Oct 26, 2016 1 / 9

Numerical monoids Definition A numerical monoid S is an additive submonoid of N with | N \ S | < ∞ . Christopher O’Neill (UC Davis) Numerical semigroups Oct 26, 2016 2 / 9

Numerical monoids Definition A numerical monoid S is an additive submonoid of N with | N \ S | < ∞ . Example McN = � 6 , 9 , 20 � = { 0 , 6 , 9 , 12 , 15 , 18 , 20 , 21 , . . . } . Christopher O’Neill (UC Davis) Numerical semigroups Oct 26, 2016 2 / 9

Numerical monoids Definition A numerical monoid S is an additive submonoid of N with | N \ S | < ∞ . Example McN = � 6 , 9 , 20 � = { 0 , 6 , 9 , 12 , 15 , 18 , 20 , 21 , . . . } . “McNugget Monoid” Christopher O’Neill (UC Davis) Numerical semigroups Oct 26, 2016 2 / 9

Numerical monoids Definition A numerical monoid S is an additive submonoid of N with | N \ S | < ∞ . Example McN = � 6 , 9 , 20 � = { 0 , 6 , 9 , 12 , 15 , 18 , 20 , 21 , . . . } . “McNugget Monoid” Factorizations: 60 = Christopher O’Neill (UC Davis) Numerical semigroups Oct 26, 2016 2 / 9

Numerical monoids Definition A numerical monoid S is an additive submonoid of N with | N \ S | < ∞ . Example McN = � 6 , 9 , 20 � = { 0 , 6 , 9 , 12 , 15 , 18 , 20 , 21 , . . . } . “McNugget Monoid” Factorizations: 60 = 7(6) + 2(9) Christopher O’Neill (UC Davis) Numerical semigroups Oct 26, 2016 2 / 9

Numerical monoids Definition A numerical monoid S is an additive submonoid of N with | N \ S | < ∞ . Example McN = � 6 , 9 , 20 � = { 0 , 6 , 9 , 12 , 15 , 18 , 20 , 21 , . . . } . “McNugget Monoid” Factorizations: 60 = 7(6) + 2(9) = 3(20) Christopher O’Neill (UC Davis) Numerical semigroups Oct 26, 2016 2 / 9

Numerical monoids Definition A numerical monoid S is an additive submonoid of N with | N \ S | < ∞ . Example McN = � 6 , 9 , 20 � = { 0 , 6 , 9 , 12 , 15 , 18 , 20 , 21 , . . . } . “McNugget Monoid” Factorizations: 60 = 7(6) + 2(9) (7 , 2 , 0) � = 3(20) (0 , 0 , 3) � Christopher O’Neill (UC Davis) Numerical semigroups Oct 26, 2016 2 / 9

Numerical monoids Definition A numerical monoid S is an additive submonoid of N with | N \ S | < ∞ . Example McN = � 6 , 9 , 20 � = { 0 , 6 , 9 , 12 , 15 , 18 , 20 , 21 , . . . } . “McNugget Monoid” Factorizations: 60 = 7(6) + 2(9) (7 , 2 , 0) � = 3(20) (0 , 0 , 3) � Unique minimal generating set: Christopher O’Neill (UC Davis) Numerical semigroups Oct 26, 2016 2 / 9

Numerical monoids Definition A numerical monoid S is an additive submonoid of N with | N \ S | < ∞ . Example McN = � 6 , 9 , 20 � = { 0 , 6 , 9 , 12 , 15 , 18 , 20 , 21 , . . . } . “McNugget Monoid” Factorizations: 60 = 7(6) + 2(9) (7 , 2 , 0) � = 3(20) (0 , 0 , 3) � Unique minimal generating set: � 15 , 17 , 22 , 32 , 40 , 42 , 56 , 58 � Christopher O’Neill (UC Davis) Numerical semigroups Oct 26, 2016 2 / 9

Numerical monoids Definition A numerical monoid S is an additive submonoid of N with | N \ S | < ∞ . Example McN = � 6 , 9 , 20 � = { 0 , 6 , 9 , 12 , 15 , 18 , 20 , 21 , . . . } . “McNugget Monoid” Factorizations: 60 = 7(6) + 2(9) (7 , 2 , 0) � = 3(20) (0 , 0 , 3) � Unique minimal generating set: � 15 , 17 , 22 , 32 , 40 , 42 , 56 , 58 � � � 15 , 17 , 22 , 40 , 42 , 58 � Christopher O’Neill (UC Davis) Numerical semigroups Oct 26, 2016 2 / 9

Numerical monoids Definition A numerical monoid S is an additive submonoid of N with | N \ S | < ∞ . Example McN = � 6 , 9 , 20 � = { 0 , 6 , 9 , 12 , 15 , 18 , 20 , 21 , . . . } . “McNugget Monoid” Factorizations: 60 = 7(6) + 2(9) (7 , 2 , 0) � = 3(20) (0 , 0 , 3) � Unique minimal generating set: � 15 , 17 , 22 , 32 , 40 , 42 , 56 , 58 � � � 15 , 17 , 22 , 40 , 42 , 58 � 32 = 15 + 17 56 = 2 · 17 + 22 Christopher O’Neill (UC Davis) Numerical semigroups Oct 26, 2016 2 / 9

Factorization invariants Fix a numerical monoid S = � n 1 , . . . , n k � . For n ∈ S , Christopher O’Neill (UC Davis) Numerical semigroups Oct 26, 2016 3 / 9

Factorization invariants Fix a numerical monoid S = � n 1 , . . . , n k � . For n ∈ S , Max and min factorization length: M( n ) , m( n ) ∈ Z ≥ 0 Christopher O’Neill (UC Davis) Numerical semigroups Oct 26, 2016 3 / 9

Factorization invariants Fix a numerical monoid S = � n 1 , . . . , n k � . For n ∈ S , Max and min factorization length: M( n ) , m( n ) ∈ Z ≥ 0 Delta set: ∆( n ) ⊂ Z ≥ 1 ∆( S ) = � m ∈ S ∆( m ) Christopher O’Neill (UC Davis) Numerical semigroups Oct 26, 2016 3 / 9

Factorization invariants Fix a numerical monoid S = � n 1 , . . . , n k � . For n ∈ S , Max and min factorization length: M( n ) , m( n ) ∈ Z ≥ 0 Delta set: ∆( n ) ⊂ Z ≥ 1 ∆( S ) = � m ∈ S ∆( m ) Catenary degree: c( n ) ∈ Z ≥ 1 c( S ) = max { c( m ) : m ∈ S } Christopher O’Neill (UC Davis) Numerical semigroups Oct 26, 2016 3 / 9

Factorization invariants Fix a numerical monoid S = � n 1 , . . . , n k � . For n ∈ S , Max and min factorization length: M( n ) , m( n ) ∈ Z ≥ 0 Delta set: ∆( n ) ⊂ Z ≥ 1 ∆( S ) = � m ∈ S ∆( m ) Catenary degree: c( n ) ∈ Z ≥ 1 c( S ) = max { c( m ) : m ∈ S } ω -primality: ω ( n ) ∈ Z ≥ 1 Christopher O’Neill (UC Davis) Numerical semigroups Oct 26, 2016 3 / 9

Factorization invariants Invariant behavior for large numerical monoid elements: 7 8 6 5 6 4 4 3 2 2 1 0 0 10 20 30 40 50 60 10 20 30 40 50 60 M : � 6 , 9 , 20 � → N m : � 6 , 9 , 20 � → N Christopher O’Neill (UC Davis) Numerical semigroups Oct 26, 2016 4 / 9

Factorization invariants Invariant behavior for large numerical monoid elements: 4 3 2 1 50 100 150 ∆ : � 6 , 9 , 20 � → 2 N Christopher O’Neill (UC Davis) Numerical semigroups Oct 26, 2016 5 / 9

5 10 50 100 150 200 250 0 15 Factorization invariants Invariant behavior for large numerical monoid elements: n �− → c( � n , n + 6 , n + 9 , n + 20 � ) Christopher O’Neill (UC Davis) Numerical semigroups Oct 26, 2016 6 / 9

Software! Christopher O’Neill (UC Davis) Numerical semigroups Oct 26, 2016 7 / 9

Software! Input: Numerical monoid S = � n 1 , . . . , n k � Monoid element n ∈ S Christopher O’Neill (UC Davis) Numerical semigroups Oct 26, 2016 7 / 9

Software! Input: Numerical monoid S = � n 1 , . . . , n k � Monoid element n ∈ S Output: Invariant value Delta set, catenary degree, list of factorizations, . . . Christopher O’Neill (UC Davis) Numerical semigroups Oct 26, 2016 7 / 9

Software! Input: Numerical monoid S = � n 1 , . . . , n k � Monoid element n ∈ S Output: Invariant value Delta set, catenary degree, list of factorizations, . . . GAP Numerical Semigroups Package, available at http://www.gap-system.org/Packages/numericalsgps.html . Christopher O’Neill (UC Davis) Numerical semigroups Oct 26, 2016 7 / 9

Software! Input: Numerical monoid S = � n 1 , . . . , n k � Monoid element n ∈ S Output: Invariant value Delta set, catenary degree, list of factorizations, . . . GAP Numerical Semigroups Package, available at http://www.gap-system.org/Packages/numericalsgps.html . Let’s see it in action! Christopher O’Neill (UC Davis) Numerical semigroups Oct 26, 2016 7 / 9

50 100 150 200 50 100 150 200 GAP from within Sage Using GAP within Sage via gap console() Christopher O’Neill (UC Davis) Numerical semigroups Oct 26, 2016 8 / 9

50 100 150 200 50 100 150 200 GAP from within Sage Using GAP within Sage via gap console() Conflated message Christopher O’Neill (UC Davis) Numerical semigroups Oct 26, 2016 8 / 9

50 100 150 200 50 100 150 200 GAP from within Sage Using GAP within Sage via gap console() Conflated message Less intuitive language Christopher O’Neill (UC Davis) Numerical semigroups Oct 26, 2016 8 / 9

200 50 200 150 100 50 100 150 GAP from within Sage Using GAP within Sage via gap console() Conflated message Less intuitive language Port data to Sage for plots: Christopher O’Neill (UC Davis) Numerical semigroups Oct 26, 2016 8 / 9

200 150 200 150 100 50 50 100 GAP from within Sage Using GAP within Sage via gap console() Conflated message Less intuitive language Port data to Sage for plots: Conjecture: For all S , s ∈ N \ S , there exists a height 2 Leamer s -atom. Christopher O’Neill (UC Davis) Numerical semigroups Oct 26, 2016 8 / 9

200 150 200 150 100 50 50 100 GAP from within Sage Using GAP within Sage via gap console() Conflated message Less intuitive language Port data to Sage for plots: Conjecture: For all S , s ∈ N \ S , there exists a height 2 Leamer s -atom. S = � 14 , 17 � Christopher O’Neill (UC Davis) Numerical semigroups Oct 26, 2016 8 / 9

Sage wrapper Solution: NumericalSemigroup Sage class! Christopher O’Neill (UC Davis) Numerical semigroups Oct 26, 2016 9 / 9

Sage wrapper Solution: NumericalSemigroup Sage class! NumericalSemigroup instance � GAP object Christopher O’Neill (UC Davis) Numerical semigroups Oct 26, 2016 9 / 9