Exploring the Catenary Degrees of Singular Arithmetical Congruence Monoids

Transcription

1 Exploring the Catenary Degrees of Singular Arithmetical Congruence Monoids Scott Chapman Sam Houston State University November 13, 2016 Chapman (Sam Houston State University) November 13, / 13

2 The Student Authors This talk is based on the work completed under my direction at the 2013 PURE REU at the University of Hawaii at Hilo by the following students. Theo McKenzie, Harvard and Sherilyn Tamagawa, Scripps College Chapman (Sam Houston State University) November 13, / 13

3 Prologue Here are some background papers for this talk. [1] A. Geroldinger and F. Halter-Koch. Congruence monoids. ACTA Arithmetica 112(2004): [2] M. Banister, M., J. Chaika, S. Chapman, and W. Meyerson. On the arithmetic of arithmetical congruence monoids. Colloq. Math 108(2007), [3] P. Baginski, and S. Chapman. Arithmetic congruence monoids: A survey. In Combinatorial and Additive Number Theory, pp Springer New York, Chapman (Sam Houston State University) November 13, / 13

4 Prologue Here are some background papers for this talk. [1] A. Geroldinger and F. Halter-Koch. Congruence monoids. ACTA Arithmetica 112(2004): [2] M. Banister, M., J. Chaika, S. Chapman, and W. Meyerson. On the arithmetic of arithmetical congruence monoids. Colloq. Math 108(2007), [3] P. Baginski, and S. Chapman. Arithmetic congruence monoids: A survey. In Combinatorial and Additive Number Theory, pp Springer New York, Chapman (Sam Houston State University) November 13, / 13

5 Prologue Here are some background papers for this talk. [1] A. Geroldinger and F. Halter-Koch. Congruence monoids. ACTA Arithmetica 112(2004): [2] M. Banister, M., J. Chaika, S. Chapman, and W. Meyerson. On the arithmetic of arithmetical congruence monoids. Colloq. Math 108(2007), [3] P. Baginski, and S. Chapman. Arithmetic congruence monoids: A survey. In Combinatorial and Additive Number Theory, pp Springer New York, Chapman (Sam Houston State University) November 13, / 13

6 Prologue Here are some background papers for this talk. [1] A. Geroldinger and F. Halter-Koch. Congruence monoids. ACTA Arithmetica 112(2004): [2] M. Banister, M., J. Chaika, S. Chapman, and W. Meyerson. On the arithmetic of arithmetical congruence monoids. Colloq. Math 108(2007), [3] P. Baginski, and S. Chapman. Arithmetic congruence monoids: A survey. In Combinatorial and Additive Number Theory, pp Springer New York, Chapman (Sam Houston State University) November 13, / 13

7 Motivation THIS IS A COMMUTATIVE ALGEBRA TALK? KIND OF. Theorem Z = {... 3, 2, 1, 0, 1, 2, 3,...} is a unique factorization domain (UFD). Theorem The multiplicative monoid Z = {... 3, 2, 1, 1, 2, 3,...} is a unique factorization monoid (UFM). Chapman (Sam Houston State University) November 13, / 13

8 Motivation THIS IS A COMMUTATIVE ALGEBRA TALK? KIND OF. Theorem Z = {... 3, 2, 1, 0, 1, 2, 3,...} is a unique factorization domain (UFD). Theorem The multiplicative monoid Z = {... 3, 2, 1, 1, 2, 3,...} is a unique factorization monoid (UFM). Chapman (Sam Houston State University) November 13, / 13

9 Motivation THIS IS A COMMUTATIVE ALGEBRA TALK? KIND OF. Theorem Z = {... 3, 2, 1, 0, 1, 2, 3,...} is a unique factorization domain (UFD). Theorem The multiplicative monoid Z = {... 3, 2, 1, 1, 2, 3,...} is a unique factorization monoid (UFM). Chapman (Sam Houston State University) November 13, / 13

10 Motivation THIS IS A COMMUTATIVE ALGEBRA TALK? KIND OF. Theorem Z = {... 3, 2, 1, 0, 1, 2, 3,...} is a unique factorization domain (UFD). Theorem The multiplicative monoid Z = {... 3, 2, 1, 1, 2, 3,...} is a unique factorization monoid (UFM). Chapman (Sam Houston State University) November 13, / 13

11 Our Object of Interest Definition Let Γ Z n = {0, 1,..., n 1} be a multiplicatively closed subset. The subset Z Γ = {n Z n x (mod n) for some x Γ} {1}. is a multiplicatively closed subset of Z known as a congruence monoid (of modulus n). Chapman (Sam Houston State University) November 13, / 13

12 A Classic Result Theorem (James & Niven, 1954 PAMS) Let n 2 be a positive integer. A congruence monoid S Γ is a UFM if and only if Γ = {x gcd(x, n) = 1}. An alternate proof of this result can be found in Halter-Koch (Semigroup Forum 1991) which uses the notion of a divisor theory divisor theory Krull domains. Chapman (Sam Houston State University) November 13, / 13

13 A Classic Result Theorem (James & Niven, 1954 PAMS) Let n 2 be a positive integer. A congruence monoid S Γ is a UFM if and only if Γ = {x gcd(x, n) = 1}. An alternate proof of this result can be found in Halter-Koch (Semigroup Forum 1991) which uses the notion of a divisor theory divisor theory Krull domains. Chapman (Sam Houston State University) November 13, / 13

14 A Classic Result Theorem (James & Niven, 1954 PAMS) Let n 2 be a positive integer. A congruence monoid S Γ is a UFM if and only if Γ = {x gcd(x, n) = 1}. An alternate proof of this result can be found in Halter-Koch (Semigroup Forum 1991) which uses the notion of a divisor theory divisor theory Krull domains. Chapman (Sam Houston State University) November 13, / 13

15 The Other Extreme Theorem If Γ = {x gcd(x, n) 1}, then Z Γ is a half-factorial monoid. Half-factorial means that if α 1 α s = β 1 β t where each α i and β j is irreducible, then s = t. Chapman (Sam Houston State University) November 13, / 13

16 The Other Extreme Theorem If Γ = {x gcd(x, n) 1}, then Z Γ is a half-factorial monoid. Half-factorial means that if α 1 α s = β 1 β t where each α i and β j is irreducible, then s = t. Chapman (Sam Houston State University) November 13, / 13

17 A Special Case If Γ = {x}, then Z Γ is called an arithmetical congruence monoid (or ACM). Hence, an ACM is merely an arithmetic sequence that is closed under multiplication. Example is known as the Hilbert Monoid. Example is known as Myerson s monoid 1, 5, 9, 13, 17, 21,... = 1 + 4N 0 1, 4, 10, 16, 22,... = 4 + 6N 0 {1} Chapman (Sam Houston State University) November 13, / 13

18 A Special Case If Γ = {x}, then Z Γ is called an arithmetical congruence monoid (or ACM). Hence, an ACM is merely an arithmetic sequence that is closed under multiplication. Example is known as the Hilbert Monoid. Example is known as Myerson s monoid 1, 5, 9, 13, 17, 21,... = 1 + 4N 0 1, 4, 10, 16, 22,... = 4 + 6N 0 {1} Chapman (Sam Houston State University) November 13, / 13

19 A Special Case If Γ = {x}, then Z Γ is called an arithmetical congruence monoid (or ACM). Hence, an ACM is merely an arithmetic sequence that is closed under multiplication. Example is known as the Hilbert Monoid. Example is known as Myerson s monoid 1, 5, 9, 13, 17, 21,... = 1 + 4N 0 1, 4, 10, 16, 22,... = 4 + 6N 0 {1} Chapman (Sam Houston State University) November 13, / 13

20 A Special Case If Γ = {x}, then Z Γ is called an arithmetical congruence monoid (or ACM). Hence, an ACM is merely an arithmetic sequence that is closed under multiplication. Example is known as the Hilbert Monoid. Example is known as Myerson s monoid 1, 5, 9, 13, 17, 21,... = 1 + 4N 0 1, 4, 10, 16, 22,... = 4 + 6N 0 {1} Chapman (Sam Houston State University) November 13, / 13

21 Types of ACMs Thus every ACM can be written uniquely in the form M a,b = a + bn 0 {1} and they break into three catagories. Regular ACMs: These correspond to a = 1. Singlar ACMs: These come in two types: Local: gcd (a, b) = p n for some prime p; Global: gcd (a, b) = d > 1 and d is composite and not a power of a prime. Chapman (Sam Houston State University) November 13, / 13

22 Types of ACMs Thus every ACM can be written uniquely in the form M a,b = a + bn 0 {1} and they break into three catagories. Regular ACMs: These correspond to a = 1. Singlar ACMs: These come in two types: Local: gcd (a, b) = p n for some prime p; Global: gcd (a, b) = d > 1 and d is composite and not a power of a prime. Chapman (Sam Houston State University) November 13, / 13

23 Types of ACMs Thus every ACM can be written uniquely in the form M a,b = a + bn 0 {1} and they break into three catagories. Regular ACMs: These correspond to a = 1. Singlar ACMs: These come in two types: Local: gcd (a, b) = p n for some prime p; Global: gcd (a, b) = d > 1 and d is composite and not a power of a prime. Chapman (Sam Houston State University) November 13, / 13

24 Types of ACMs Thus every ACM can be written uniquely in the form M a,b = a + bn 0 {1} and they break into three catagories. Regular ACMs: These correspond to a = 1. Singlar ACMs: These come in two types: Local: gcd (a, b) = p n for some prime p; Global: gcd (a, b) = d > 1 and d is composite and not a power of a prime. Chapman (Sam Houston State University) November 13, / 13

25 Types of ACMs Thus every ACM can be written uniquely in the form M a,b = a + bn 0 {1} and they break into three catagories. Regular ACMs: These correspond to a = 1. Singlar ACMs: These come in two types: Local: gcd (a, b) = p n for some prime p; Global: gcd (a, b) = d > 1 and d is composite and not a power of a prime. Chapman (Sam Houston State University) November 13, / 13

26 The Regular Case If a = 1, then the factorization properties of M 1,b are identical to those of the multiplicative monoid of a Dedekind domain D with class group Z b where each divisor class of the class group contains a nonzero prime ideal. For instance, ρ(m 1,b ) = D(Z n ) 2. Moreover, if M a,b is global, then ρ(m a,b ) =. If M a,b is local, then ρ(m a,b ) = n+k 1 k where n is the smallest integer such that p n M a,b. Chapman (Sam Houston State University) November 13, / 13

27 The Regular Case If a = 1, then the factorization properties of M 1,b are identical to those of the multiplicative monoid of a Dedekind domain D with class group Z b where each divisor class of the class group contains a nonzero prime ideal. For instance, ρ(m 1,b ) = D(Z n ) 2. Moreover, if M a,b is global, then ρ(m a,b ) =. If M a,b is local, then ρ(m a,b ) = n+k 1 k where n is the smallest integer such that p n M a,b. Chapman (Sam Houston State University) November 13, / 13

28 The Regular Case If a = 1, then the factorization properties of M 1,b are identical to those of the multiplicative monoid of a Dedekind domain D with class group Z b where each divisor class of the class group contains a nonzero prime ideal. For instance, ρ(m 1,b ) = D(Z n ) 2. Moreover, if M a,b is global, then ρ(m a,b ) =. If M a,b is local, then ρ(m a,b ) = n+k 1 k where n is the smallest integer such that p n M a,b. Chapman (Sam Houston State University) November 13, / 13

29 The Regular Case If a = 1, then the factorization properties of M 1,b are identical to those of the multiplicative monoid of a Dedekind domain D with class group Z b where each divisor class of the class group contains a nonzero prime ideal. For instance, ρ(m 1,b ) = D(Z n ) 2. Moreover, if M a,b is global, then ρ(m a,b ) =. If M a,b is local, then ρ(m a,b ) = n+k 1 k where n is the smallest integer such that p n M a,b. Chapman (Sam Houston State University) November 13, / 13

30 Definitions Suppose that M is reduced, x M is not the identity, and that F : x = α 1 α n β 1 β s and F : x = α 1 α n γ 1 γ t are distinct atomic factorizations such that β i γ j for all i, j. With notation as above, we define gcd(f, F ) = α 1 α n and the distance between F and F by d(f, F ) = max{s, t}. Extend d to all pairs of factorizations by d(f, F ) = 0. Chapman (Sam Houston State University) November 13, / 13

31 Definitions Suppose that M is reduced, x M is not the identity, and that F : x = α 1 α n β 1 β s and F : x = α 1 α n γ 1 γ t are distinct atomic factorizations such that β i γ j for all i, j. With notation as above, we define gcd(f, F ) = α 1 α n and the distance between F and F by d(f, F ) = max{s, t}. Extend d to all pairs of factorizations by d(f, F ) = 0. Chapman (Sam Houston State University) November 13, / 13

32 The Distance is Amazing The distance function acts as a metric. The following for a numerical monoid can easily be shown (and are in fact true in general). Theorem Let F 1, F 2 and F 3 be factorizations of x in a numerical monoid S. 1. d(f 1, F 2 ) = 0 if and only if F 1 = F d(f 1, F 2 ) = d(f 2, F 1 ). 3. d(f 1, F 2 ) d(f 1, F 3 ) + d(f 3, F 2 ). 4. d(f 3 F 1, F 3 F 2 ) = d(f 1, F 2 ). 5. d(f k 1, F k 2 ) = kd(f 1, F 2 ). Chapman (Sam Houston State University) November 13, / 13

33 The Distance is Amazing The distance function acts as a metric. The following for a numerical monoid can easily be shown (and are in fact true in general). Theorem Let F 1, F 2 and F 3 be factorizations of x in a numerical monoid S. 1. d(f 1, F 2 ) = 0 if and only if F 1 = F d(f 1, F 2 ) = d(f 2, F 1 ). 3. d(f 1, F 2 ) d(f 1, F 3 ) + d(f 3, F 2 ). 4. d(f 3 F 1, F 3 F 2 ) = d(f 1, F 2 ). 5. d(f k 1, F k 2 ) = kd(f 1, F 2 ). Chapman (Sam Houston State University) November 13, / 13

34 More Definitions An N-chain of factorizations from F to F is a sequence F 0,..., F k such that each F i is a factorization of x, F 0 = F and F k = F, and d(f i, F i+1 ) N for all i < k. The catenary degree of x, denoted c(x), is the least N Z 0 { } such that for any two factorizations F, F of x there is an N-chain between F and F. The catenary degree of the monoid M is defined as c(m) = sup{ c(x) x M \ M }. Note: If S does not have unique factorization, then c(s) 2 and if c(s) = 2, then S is half-factorial. Chapman (Sam Houston State University) November 13, / 13

35 More Definitions An N-chain of factorizations from F to F is a sequence F 0,..., F k such that each F i is a factorization of x, F 0 = F and F k = F, and d(f i, F i+1 ) N for all i < k. The catenary degree of x, denoted c(x), is the least N Z 0 { } such that for any two factorizations F, F of x there is an N-chain between F and F. The catenary degree of the monoid M is defined as c(m) = sup{ c(x) x M \ M }. Note: If S does not have unique factorization, then c(s) 2 and if c(s) = 2, then S is half-factorial. Chapman (Sam Houston State University) November 13, / 13

36 More Definitions An N-chain of factorizations from F to F is a sequence F 0,..., F k such that each F i is a factorization of x, F 0 = F and F k = F, and d(f i, F i+1 ) N for all i < k. The catenary degree of x, denoted c(x), is the least N Z 0 { } such that for any two factorizations F, F of x there is an N-chain between F and F. The catenary degree of the monoid M is defined as c(m) = sup{ c(x) x M \ M }. Note: If S does not have unique factorization, then c(s) 2 and if c(s) = 2, then S is half-factorial. Chapman (Sam Houston State University) November 13, / 13

37 More Definitions An N-chain of factorizations from F to F is a sequence F 0,..., F k such that each F i is a factorization of x, F 0 = F and F k = F, and d(f i, F i+1 ) N for all i < k. The catenary degree of x, denoted c(x), is the least N Z 0 { } such that for any two factorizations F, F of x there is an N-chain between F and F. The catenary degree of the monoid M is defined as c(m) = sup{ c(x) x M \ M }. Note: If S does not have unique factorization, then c(s) 2 and if c(s) = 2, then S is half-factorial. Chapman (Sam Houston State University) November 13, / 13