Numerical semigroups: a sales pitch Christopher ONeill San Diego - PowerPoint PPT Presentation

Numerical semigroups: a sales pitch Christopher O’Neill San Diego State University cdoneill@sdsu.edu Slides available: http://www.tinyurl.com/JMM2020-FURM January 17, 2020 Christopher O’Neill (SDSU) Numerical semigroup: a sales pitch January 17, 2020 1 / 14

Numerical semigroups Definition A numerical semigroup S ⊂ Z ≥ 0 : closed under addition . Christopher O’Neill (SDSU) Numerical semigroup: a sales pitch January 17, 2020 2 / 14

Numerical semigroups Definition A numerical semigroup S ⊂ Z ≥ 0 : closed under addition . Example McN = � 6 , 9 , 20 � = { 0 , 6 , 9 , 12 , 15 , 18 , 20 , . . . } . Christopher O’Neill (SDSU) Numerical semigroup: a sales pitch January 17, 2020 2 / 14

Numerical semigroups Definition A numerical semigroup S ⊂ Z ≥ 0 : closed under addition . Example McN = � 6 , 9 , 20 � = { 0 , 6 , 9 , 12 , 15 , 18 , 20 , . . . } . “McNugget Semigroup” Christopher O’Neill (SDSU) Numerical semigroup: a sales pitch January 17, 2020 2 / 14

Numerical semigroups Definition A numerical semigroup S ⊂ Z ≥ 0 : closed under addition . Example McN = � 6 , 9 , 20 � = { 0 , 6 , 9 , 12 , 15 , 18 , 20 , . . . } . “McNugget Semigroup” Factorizations: 60 = Christopher O’Neill (SDSU) Numerical semigroup: a sales pitch January 17, 2020 2 / 14

Numerical semigroups Definition A numerical semigroup S ⊂ Z ≥ 0 : closed under addition . Example McN = � 6 , 9 , 20 � = { 0 , 6 , 9 , 12 , 15 , 18 , 20 , . . . } . “McNugget Semigroup” Factorizations: 60 = 7(6) + 2(9) Christopher O’Neill (SDSU) Numerical semigroup: a sales pitch January 17, 2020 2 / 14

Numerical semigroups Definition A numerical semigroup S ⊂ Z ≥ 0 : closed under addition . Example McN = � 6 , 9 , 20 � = { 0 , 6 , 9 , 12 , 15 , 18 , 20 , . . . } . “McNugget Semigroup” Factorizations: 60 = 7(6) + 2(9) = 3(20) Christopher O’Neill (SDSU) Numerical semigroup: a sales pitch January 17, 2020 2 / 14

Numerical semigroups Definition A numerical semigroup S ⊂ Z ≥ 0 : closed under addition . Example McN = � 6 , 9 , 20 � = { 0 , 6 , 9 , 12 , 15 , 18 , 20 , . . . } . “McNugget Semigroup” Factorizations: 60 = 7(6) + 2(9) (7 , 2 , 0) � = 3(20) (0 , 0 , 3) � Christopher O’Neill (SDSU) Numerical semigroup: a sales pitch January 17, 2020 2 / 14

Extremal factorization length Fix a numerical semigroup S = � n 1 , . . . , n k � and an element n ∈ S . Christopher O’Neill (SDSU) Numerical semigroup: a sales pitch January 17, 2020 3 / 14

Extremal factorization length Fix a numerical semigroup S = � n 1 , . . . , n k � and an element n ∈ S . A factorization a = ( a 1 , . . . , a k ) ∈ Z k ≥ 0 of n n = a 1 n 1 + · · · + a k n k has length | a | = a 1 + · · · + a k . Christopher O’Neill (SDSU) Numerical semigroup: a sales pitch January 17, 2020 3 / 14

Extremal factorization length Fix a numerical semigroup S = � n 1 , . . . , n k � and an element n ∈ S . A factorization a = ( a 1 , . . . , a k ) ∈ Z k ≥ 0 of n n = a 1 n 1 + · · · + a k n k has length | a | = a 1 + · · · + a k . Example All factorizations of 60 ∈ � 6 , 9 , 20 � : (10 , 0 , 0) , (7 , 2 , 0) , (4 , 4 , 0) , (1 , 6 , 0) , (0 , 0 , 3) Christopher O’Neill (SDSU) Numerical semigroup: a sales pitch January 17, 2020 3 / 14

Extremal factorization length Fix a numerical semigroup S = � n 1 , . . . , n k � and an element n ∈ S . A factorization a = ( a 1 , . . . , a k ) ∈ Z k ≥ 0 of n n = a 1 n 1 + · · · + a k n k has length | a | = a 1 + · · · + a k . Example All factorizations of 60 ∈ � 6 , 9 , 20 � : (10 , 0 , 0) , (7 , 2 , 0) , (4 , 4 , 0) , (1 , 6 , 0) , (0 , 0 , 3) Lengths: 3 , 7 , 8 , 9 , 10. Christopher O’Neill (SDSU) Numerical semigroup: a sales pitch January 17, 2020 3 / 14

Extremal factorization length Fix a numerical semigroup S = � n 1 , . . . , n k � and an element n ∈ S . A factorization a = ( a 1 , . . . , a k ) ∈ Z k ≥ 0 of n n = a 1 n 1 + · · · + a k n k has length | a | = a 1 + · · · + a k . Example All factorizations of 60 ∈ � 6 , 9 , 20 � : (10 , 0 , 0) , (7 , 2 , 0) , (4 , 4 , 0) , (1 , 6 , 0) , (0 , 0 , 3) Lengths: 3 , 7 , 8 , 9 , 10. All factorizations of 1000001: Christopher O’Neill (SDSU) Numerical semigroup: a sales pitch January 17, 2020 3 / 14

Extremal factorization length Fix a numerical semigroup S = � n 1 , . . . , n k � and an element n ∈ S . A factorization a = ( a 1 , . . . , a k ) ∈ Z k ≥ 0 of n n = a 1 n 1 + · · · + a k n k has length | a | = a 1 + · · · + a k . Example All factorizations of 60 ∈ � 6 , 9 , 20 � : (10 , 0 , 0) , (7 , 2 , 0) , (4 , 4 , 0) , (1 , 6 , 0) , (0 , 0 , 3) Lengths: 3 , 7 , 8 , 9 , 10. All factorizations of 1000001: , . . . , � �� � � �� � shortest longest Christopher O’Neill (SDSU) Numerical semigroup: a sales pitch January 17, 2020 3 / 14

Extremal factorization length Fix a numerical semigroup S = � n 1 , . . . , n k � and an element n ∈ S . A factorization a = ( a 1 , . . . , a k ) ∈ Z k ≥ 0 of n n = a 1 n 1 + · · · + a k n k has length | a | = a 1 + · · · + a k . Example All factorizations of 60 ∈ � 6 , 9 , 20 � : (10 , 0 , 0) , (7 , 2 , 0) , (4 , 4 , 0) , (1 , 6 , 0) , (0 , 0 , 3) Lengths: 3 , 7 , 8 , 9 , 10. All factorizations of 1000001: (2 , 1 , 49999) , . . . , � �� � � �� � shortest longest Christopher O’Neill (SDSU) Numerical semigroup: a sales pitch January 17, 2020 3 / 14

Extremal factorization length Fix a numerical semigroup S = � n 1 , . . . , n k � and an element n ∈ S . A factorization a = ( a 1 , . . . , a k ) ∈ Z k ≥ 0 of n n = a 1 n 1 + · · · + a k n k has length | a | = a 1 + · · · + a k . Example All factorizations of 60 ∈ � 6 , 9 , 20 � : (10 , 0 , 0) , (7 , 2 , 0) , (4 , 4 , 0) , (1 , 6 , 0) , (0 , 0 , 3) Lengths: 3 , 7 , 8 , 9 , 10. All factorizations of 1000001: (2 , 1 , 49999) , . . . , (166662 , 1 , 1) � �� � � �� � shortest longest Christopher O’Neill (SDSU) Numerical semigroup: a sales pitch January 17, 2020 3 / 14

Extremal factorization length Let S = � n 1 , . . . , n k � . For n ∈ S , let L( n ) = { a 1 + · · · + a k : n = a 1 n 1 + · · · + a k n k } denotes the length set of n Christopher O’Neill (SDSU) Numerical semigroup: a sales pitch January 17, 2020 4 / 14

Extremal factorization length Let S = � n 1 , . . . , n k � . For n ∈ S , let L( n ) = { a 1 + · · · + a k : n = a 1 n 1 + · · · + a k n k } denotes the length set of n , and M( n ) = max L( n ) and m( n ) = min L( n ) denote the maximum and minimum factorization lengths of n . Christopher O’Neill (SDSU) Numerical semigroup: a sales pitch January 17, 2020 4 / 14

Extremal factorization length Let S = � n 1 , . . . , n k � . For n ∈ S , let L( n ) = { a 1 + · · · + a k : n = a 1 n 1 + · · · + a k n k } denotes the length set of n , and M( n ) = max L( n ) and m( n ) = min L( n ) denote the maximum and minimum factorization lengths of n . Observations Max length factorization: lots of small generators Min length factorization: lots of large generators Christopher O’Neill (SDSU) Numerical semigroup: a sales pitch January 17, 2020 4 / 14

Extremal factorization length Let S = � n 1 , . . . , n k � . For n ∈ S , let L( n ) = { a 1 + · · · + a k : n = a 1 n 1 + · · · + a k n k } denotes the length set of n , and M( n ) = max L( n ) and m( n ) = min L( n ) denote the maximum and minimum factorization lengths of n . Observations Max length factorization: lots of small generators Min length factorization: lots of large generators Example S = � 5 , 16 , 17 , 18 , 19 � : Christopher O’Neill (SDSU) Numerical semigroup: a sales pitch January 17, 2020 4 / 14

Extremal factorization length Let S = � n 1 , . . . , n k � . For n ∈ S , let L( n ) = { a 1 + · · · + a k : n = a 1 n 1 + · · · + a k n k } denotes the length set of n , and M( n ) = max L( n ) and m( n ) = min L( n ) denote the maximum and minimum factorization lengths of n . Observations Max length factorization: lots of small generators Min length factorization: lots of large generators Example S = � 5 , 16 , 17 , 18 , 19 � : m(82) = 5 with 82 = 3(16) + 2(17) m(462) = 25 with 462 = 3(16) + 2(17) + 20(19) Christopher O’Neill (SDSU) Numerical semigroup: a sales pitch January 17, 2020 4 / 14

12 10 8 6 4 2 10 20 30 40 50 60 70 80 Extremal factorization length Let S = � n 1 , . . . , n k � . For n ∈ S , let M( n ) = max L( n ) and m( n ) = min L( n ). Christopher O’Neill (SDSU) Numerical semigroup: a sales pitch January 17, 2020 5 / 14

12 10 8 6 4 2 10 20 30 40 50 60 70 80 Extremal factorization length Let S = � n 1 , . . . , n k � . For n ∈ S , let M( n ) = max L( n ) and m( n ) = min L( n ). Example: max length in S = � 6 , 9 , 20 � Christopher O’Neill (SDSU) Numerical semigroup: a sales pitch January 17, 2020 5 / 14

Extremal factorization length Let S = � n 1 , . . . , n k � . For n ∈ S , let M( n ) = max L( n ) and m( n ) = min L( n ). Example: max length in S = � 6 , 9 , 20 � 12 10 8 6 4 2 10 20 30 40 50 60 70 80 M( n ) : S → N Christopher O’Neill (SDSU) Numerical semigroup: a sales pitch January 17, 2020 5 / 14