- Email: [email protected]
Factorization and catenary degree in 3-generated numerical semigroups
Electronic Notes in Discrete Mathematics 34 (2009) 157–161 www.elsevier.com/locate/endm
Factorization and catenary degree in 3-generated numerical semigroups 1 Francesc Aguil´o-Gost 2 Departament de Matem` atica Aplicada IV Universitat Polit`ecnica de Catalunya Barcelona, Spain
Pedro A. Garc´ıa-S´anchez 3 ´ Departamento de Algebra Universidad de Granada Granada, Spain
Abstract Given a numerical semigroup S(A), generated by A = {a, b, N } ⊂ N with 0 < a < b < N and gcd(a, b, N ) = 1, we give a parameterization of the set F(m; A) = {(x, y, z) ∈ N3 | xa + yb + zN = m} for any m ∈ S(A). We also give the catenary degree of S(A), c(A). Boths results need the computation of an L-shaped tile, related to the set A, that has time-complexity O(log N ) in the worst case. Keywords: Double-loop digraph, numerical semigroup, L-shape, factorization, catenary degree.
1
Work supported by the projects MTM2008-06620-C03-01/MTM, DURSI 2005SGR00256, MTM2007-62346 and FEDER funds 2 Email: [email protected] 3 Email: [email protected]
1571-0653/$ – see front matter © 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.endm.2009.07.026
158
1
F. Aguiló-Gost, P.A. García-Sánchez / Electronic Notes in Discrete Mathematics 34 (2009) 157–161
Introduction
Given a set A = {a, b, N } with gcd(a, b, N ) = 1, set S(A) = {αa + βb + γN | α, β, γ ∈ N} the numerical semigroup generated by A. Set F (m; A) = {(x, y, z) ∈ N3 | xa + yb + zN = m} the set of factorizations of m in S(A) and d(m; A) = |F (m; A)| the denumerant of m in S(A). For every m ∈ S(A), Ap(S(A), m) = {s ∈ S(A) | s − m ∈ S(A)} is the Ap´ery set of m associated to S(A). See [7]. Every ∈ Z3 can be uniquely expressed as = + − − , with + , − ∈ N3 and + · − = 0 (the dot product). Define the norm of as = max{|+ |, |− |}, where |(x, y, z)| = x+y+z for (x, y, z) ∈ N3 . Define dist(, ) = − , the distance between and for , ∈ N3 . Given m ∈ S(A) and , ∈ F (m; A), an n-chain of factorizations from to is a sequence 0 , . . . , k ∈ F (m; A) such that 0 = , k = and dist( i , i+1 ) ≤ n for all i. The catenary degree of m, c(m), is the minimal n ∈ N ∪ {∞} such that for any two factorizations , ∈ F (m; A), there is an n-chain from to . The catenary degree of S(A), c(A), is defined by c(A) = sup{c(m) | m ∈ S(A)}. See [3]. We denote the equivalence class of m ∈ Z modulo N ∈ N by [m]N and the element mN ∈ [m]N is such that mN ∈ {0, 1, ..., N − 1}. A weighted double-loop digraph, G = G(N; a, b; a, b) = G(V, E), is a directed graph with a set of vertices V = ZN and set of weighted directed edges E = {mN → b (m + a)N , mN → (m + b)N | mN ∈ V }. The values a and b are the steps of G. The weights are given by a and b for arcs defined by the steps a and b, respectively. The (directed) distance d(v1 , v2 ) between the vertices v1 and v2 in G is defined as usual. It has been shown that each double-loop digraph G has linked a minimum distance diagram that encodes de distance from 0 to any other vertex.
y w
h
v = (−w, h)
l u = (l, −y)
These diagrams have the form of an L-shape and are denoted by their lengths H = L(l, h, w, y) ([6,4]). The area of H is lh − wy = N and the lengths can be computed in O(log N) ([8]). They periodically tessellate the plane through the vectors u = (l, −y) and v = (−w, h) and the following
F. Aguiló-Gost, P.A. García-Sánchez / Electronic Notes in Discrete Mathematics 34 (2009) 157–161
159
relations are fulfilled la − yb ≡ 0 (mod N),
(1)
−wa + hb ≡ 0 (mod N).
The minimum distance diagram H associated to G(N ; a, b; a, b) encodes the set Ap(S(A), N) in the following sense: Each unit square with coordinates (x0 , y0 ) inside H represents the element x0 a + y0 b ∈ Ap(S(A), N) and each element n ∈ Ap(S(A), N) can be represented by n = x0 a + y0 b and the unit square inside with coordinates (x0 , y0) is located inside H. Fixed the L-shaped H associated to G(N; a, b; a, b), there is a unique representation of this type. These coordinates can be computed in O(log N) time-complexity, in the worst case ([1]).
2
Factorizations
Definition 2.1 Let m ∈ S(A) and (x0 , y0) the coordinates of the vertex mN inside H related to G(N; a, b; a, b). Set m = x0 a + y0 b and z0 = m−m . The N (unique) basic factorization of m with respect to H is (x0 , y0, z0 ). Definition 2.2 Let A = {a, b, N } ⊂ N, with 0 < a < b < N and gcd(a, b, N ) = 1. Consider a minimum distance diagram H = L(l, h, w, y) associated to G(N ; a, b; a, b). Set δ = (la−yb)/N and θ = (hb−wa)/N. Given any m ∈ S(A) and its related basic factorization with respect H, (x0 , y0 , z0 ), the sequence of control’s parameters for F (m; A) are defined by
Sk =
8 j y +k(h−y) k 0 > > y > > > :
if δ = 0,
z0 −k(δ+θ) jδ k j k y +k(h−y) z −k(δ+θ) min{ 0 y , 0 δ }
for each k = 0, ...,
z0 δ+θ
if y = 0, if δy = 0,
Tk =
8 j x +k(l−w) k 0 > > w > > > :
z0 −k(δ+θ) jθ k j k x +k(l−w) z −k(δ+θ) min{ 0 w , 0 θ }
if θ = 0, if w = 0, if θw = 0.
.
Theorem 2.3 Let A = {a, b, N } ⊂ N, 0 < a < b < N, gcd(a, b, N ) = 1, z0 /(δ+θ) z0 /(δ+θ) m ∈ F (m; A) and δ, θ, (x0 , y0 , z0 ), {Sk }k=0 and {Tk }k=0 given in Definition 2.2. Then F(m; A) =
z0 δ+θ Sk
{(x0 + k(l − w) + sl, y0 + k(h − y) − sy, z0 − k(δ + θ) − sδ)}
k=0 s=0
∪
z0 Tk δ+θ
{(x0 + k(l − w) − tw, y0 + k(h − y) + th, z0 − k(δ + θ) − tθ)}
k=0 t=1
and d(m; A) =
z0 δ+θ
k=0
(1 + Sk + Tk ).
160
F. Aguiló-Gost, P.A. García-Sánchez / Electronic Notes in Discrete Mathematics 34 (2009) 157–161
Set F2 (m; A) = {(x, y) ∈ N2 | (x, y, z) ∈ F (m; A)}. The elements of F (m; A) are found by looking for the coordinates (x, y) ∈ F2 (m; A) from the starting point (x0 , y0 ) (part of the basic factorization of m related to H) and following the plane tessellation by H through the vectors u, v and e = u + v. The tree-like approach of F2 (87; A), A = {5, 7, 11}, is depicted in the figure; here we have H = L(5, 3, 2, 2), u = (5, −2), v = (−2, 3), e = (3, 1), δ = θ = 1, 0 ≤ k ≤ 3, {Sk }3k=0 = {0, 0, 1, 1} and {Tk }3k=0 = {1, 2, 3, 1}. Hence F (87; A) = {(2, 0, 7), (0, 3, 6), (5, 1, 5), (3, 4, 4), (1, 7, 3), (8, 2, 3), (13, 0, 2), (6, 5, 2), (4, 8, 1), (2, 11, 0), (11, 3, 1), (16, 1, 0), (9, 6, 0)}. 98 103 108 113 118 123 128 133 138 143 148 153 158 163 168 173 178 183 188 193 91 96 101 106 111 116 121 126 131 136 141 146 151 156 161 166 171 176 181 186 84 89 94 99 104 109 114 119 124 129 134 139 144 149 154 159 164 169 174 179 77 82 87 92 97 102 107 112 117 122 127 132 137 142 147 152 157 162 167 172 70 75 80 85 90 95 100 105 110 115 120 125 130 135 140 145 150 155 160 165 63 68 73 78 83 88 93 98 103 108 113 118 123 128 133 138 143 148 153 158 56 61 66 71 76 81 86 91 96 101 106 111 116 121 126 131 136 141 146 151 49 54 59 64 69 74 79 84 89 94 99 104 109 114 119 124 129 134 139 144 42 47 52 57 62 67 72 77 82 87 92 97 102 107 112 117 122 127 132 137 35 40 45 50 55 60 65 70 75 80 85 90 95 100 105 110 115 120 125 130 28 33 38 43 48 53 58 63 68 73 78 83 88 93 98 103 108 113 118 123 21 26 31 36 41 46 51 56 61 66 71 76 81 86 91 96 101 106 111 116 14 19 24 29 34 39 44 49 54 59 64 69 74 79 84 89 94 99 104 109 7 12 17 22 27 32 37 42 47 52 57 62 67 72 77 82 87 92 97 102 0
3
5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95
Catenary Degree
From now on let us assume A = {a, b, N }, 0 < a < b < N, gcd(a, b, N ) = 1, G(N ; a, b; a, b) has the related minimum distance diagram H = L(l, h, w, y), δ and θ are given in Definition 2.2. Set α = (l, −y, −δ), β = (−w, h, −θ) and γ = α + β = (l − w, h − y, −(δ + θ)). Lemma 3.1 α = l, β = max{h, w + θ}, γ = l − w + h − y. From the tree like shape described in Theorem 2.3 for the set F (m; A), with m ∈ S(A), one easily deduces the following consequence. Corollary 3.2 c(A) ≤ max{l, w + θ, h, l − w + h − y}. The idea of the proof of the following theorem is based on the concept of R-classes. See [2] for more details. Theorem 3.3 •
If w = 0, then c(A) = max{l, h}.
F. Aguiló-Gost, P.A. García-Sánchez / Electronic Notes in Discrete Mathematics 34 (2009) 157–161 •
If y = 0, then
•
If wy = 0, then
161
⎧ ⎨ max{w, l + l (h − w)} if θ = 0, w c(A) = ⎩ max{l, w + θ, h} otherwise.
c(A) =
⎧ ⎪ ⎪ max{l, l − w + h − y} ⎪ ⎨
if δ = 0,
max{w, l − y + wl (h − w)} if θ = 0, ⎪ ⎪ ⎪ ⎩ max{l, w + θ, h, l − w + h − y} otherwise.
References [1] Aguil´ o-Gost, F. and J. Barguilla, Computing coordinates inside an L-shaped tile, VI Jornadas de Matem´atica Discreta y Algor´ıtmica, Ed. Josep Conde et al., Edicions i Publicacions de la UdL, Universitat de Lleida, Spain (2008), 35–41. [2] Chapman, S. T., P. A. Garc´ıa-S´ anchez, D. Llena, V. Ponomarenko, and J. C. Rosales, The catenary and tame degree in finitely generated cancellative monoids, Manuscripta Math. 120 (2006), 253–264. [3] Chapman, S. T., P. A. Garc´ıa-S´ anchez, D. Llena, The catenary and tame degree of a numerical semigroup, Forum Math. 21 (2009), 117–129. [4] Cheng Y. and F. Hwang, Diameters of Weighted Double-Loop Networks, J. Algorithms 9 (1988), 401–410. [5] Delgado, M., P. A. Garc´ıa-S´ anchez and J. Morais, “numericalsgps”: a GAP [9] package on numerical semigroups. (http://www.gap-system.org/Packages/numericalsgps.html). [6] Fiol, M.A., J.L.A. Yebra, I. Alegre and M. Valero, A discrete optimization problem in local networks and data alignment, IEEE Trans. Comput. C-36 (1987), 702–713. [7] Ram´ırez Alfons´ın, J.L., “The Diophantine Frobenius Problem”. Oxford Univ. Press (2005) Oxford. ISBN 0-19-856820-7 978-0-19-856820-9. ¨ [8] R¨ odseth, Oystein J., On a linear Diophantine problem of Frobenius, J. Reine Angew. Math. 301 (1978) 171–178. [9] The GAP Group. GAP – Groups, Algorithms, and Programming, Version 4.4, 2004. (http://www.gap-system.org).
Factorization and catenary degree in 3-generated numerical semigroups 1 Francesc Aguil´o-Gost 2 Departament de Matem` atica Aplicada IV Universitat Polit`ecnica de Catalunya Barcelona, Spain
Pedro A. Garc´ıa-S´anchez 3 ´ Departamento de Algebra Universidad de Granada Granada, Spain
Abstract Given a numerical semigroup S(A), generated by A = {a, b, N } ⊂ N with 0 < a < b < N and gcd(a, b, N ) = 1, we give a parameterization of the set F(m; A) = {(x, y, z) ∈ N3 | xa + yb + zN = m} for any m ∈ S(A). We also give the catenary degree of S(A), c(A). Boths results need the computation of an L-shaped tile, related to the set A, that has time-complexity O(log N ) in the worst case. Keywords: Double-loop digraph, numerical semigroup, L-shape, factorization, catenary degree.
1
Work supported by the projects MTM2008-06620-C03-01/MTM, DURSI 2005SGR00256, MTM2007-62346 and FEDER funds 2 Email: [email protected] 3 Email: [email protected]
1571-0653/$ – see front matter © 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.endm.2009.07.026
158
1
F. Aguiló-Gost, P.A. García-Sánchez / Electronic Notes in Discrete Mathematics 34 (2009) 157–161
Introduction
Given a set A = {a, b, N } with gcd(a, b, N ) = 1, set S(A) = {αa + βb + γN | α, β, γ ∈ N} the numerical semigroup generated by A. Set F (m; A) = {(x, y, z) ∈ N3 | xa + yb + zN = m} the set of factorizations of m in S(A) and d(m; A) = |F (m; A)| the denumerant of m in S(A). For every m ∈ S(A), Ap(S(A), m) = {s ∈ S(A) | s − m ∈ S(A)} is the Ap´ery set of m associated to S(A). See [7]. Every ∈ Z3 can be uniquely expressed as = + − − , with + , − ∈ N3 and + · − = 0 (the dot product). Define the norm of as = max{|+ |, |− |}, where |(x, y, z)| = x+y+z for (x, y, z) ∈ N3 . Define dist(, ) = − , the distance between and for , ∈ N3 . Given m ∈ S(A) and , ∈ F (m; A), an n-chain of factorizations from to is a sequence 0 , . . . , k ∈ F (m; A) such that 0 = , k = and dist( i , i+1 ) ≤ n for all i. The catenary degree of m, c(m), is the minimal n ∈ N ∪ {∞} such that for any two factorizations , ∈ F (m; A), there is an n-chain from to . The catenary degree of S(A), c(A), is defined by c(A) = sup{c(m) | m ∈ S(A)}. See [3]. We denote the equivalence class of m ∈ Z modulo N ∈ N by [m]N and the element mN ∈ [m]N is such that mN ∈ {0, 1, ..., N − 1}. A weighted double-loop digraph, G = G(N; a, b; a, b) = G(V, E), is a directed graph with a set of vertices V = ZN and set of weighted directed edges E = {mN → b (m + a)N , mN → (m + b)N | mN ∈ V }. The values a and b are the steps of G. The weights are given by a and b for arcs defined by the steps a and b, respectively. The (directed) distance d(v1 , v2 ) between the vertices v1 and v2 in G is defined as usual. It has been shown that each double-loop digraph G has linked a minimum distance diagram that encodes de distance from 0 to any other vertex.
y w
h
v = (−w, h)
l u = (l, −y)
These diagrams have the form of an L-shape and are denoted by their lengths H = L(l, h, w, y) ([6,4]). The area of H is lh − wy = N and the lengths can be computed in O(log N) ([8]). They periodically tessellate the plane through the vectors u = (l, −y) and v = (−w, h) and the following
F. Aguiló-Gost, P.A. García-Sánchez / Electronic Notes in Discrete Mathematics 34 (2009) 157–161
159
relations are fulfilled la − yb ≡ 0 (mod N),
(1)
−wa + hb ≡ 0 (mod N).
The minimum distance diagram H associated to G(N ; a, b; a, b) encodes the set Ap(S(A), N) in the following sense: Each unit square with coordinates (x0 , y0 ) inside H represents the element x0 a + y0 b ∈ Ap(S(A), N) and each element n ∈ Ap(S(A), N) can be represented by n = x0 a + y0 b and the unit square inside with coordinates (x0 , y0) is located inside H. Fixed the L-shaped H associated to G(N; a, b; a, b), there is a unique representation of this type. These coordinates can be computed in O(log N) time-complexity, in the worst case ([1]).
2
Factorizations
Definition 2.1 Let m ∈ S(A) and (x0 , y0) the coordinates of the vertex mN inside H related to G(N; a, b; a, b). Set m = x0 a + y0 b and z0 = m−m . The N (unique) basic factorization of m with respect to H is (x0 , y0, z0 ). Definition 2.2 Let A = {a, b, N } ⊂ N, with 0 < a < b < N and gcd(a, b, N ) = 1. Consider a minimum distance diagram H = L(l, h, w, y) associated to G(N ; a, b; a, b). Set δ = (la−yb)/N and θ = (hb−wa)/N. Given any m ∈ S(A) and its related basic factorization with respect H, (x0 , y0 , z0 ), the sequence of control’s parameters for F (m; A) are defined by
Sk =
8 j y +k(h−y) k 0 > > y >
if δ = 0,
z0 −k(δ+θ) jδ k j k y +k(h−y) z −k(δ+θ) min{ 0 y , 0 δ }
for each k = 0, ...,
z0 δ+θ
if y = 0, if δy = 0,
Tk =
8 j x +k(l−w) k 0 > > w >
z0 −k(δ+θ) jθ k j k x +k(l−w) z −k(δ+θ) min{ 0 w , 0 θ }
if θ = 0, if w = 0, if θw = 0.
.
Theorem 2.3 Let A = {a, b, N } ⊂ N, 0 < a < b < N, gcd(a, b, N ) = 1, z0 /(δ+θ) z0 /(δ+θ) m ∈ F (m; A) and δ, θ, (x0 , y0 , z0 ), {Sk }k=0 and {Tk }k=0 given in Definition 2.2. Then F(m; A) =
z0 δ+θ Sk
{(x0 + k(l − w) + sl, y0 + k(h − y) − sy, z0 − k(δ + θ) − sδ)}
k=0 s=0
∪
z0 Tk δ+θ
{(x0 + k(l − w) − tw, y0 + k(h − y) + th, z0 − k(δ + θ) − tθ)}
k=0 t=1
and d(m; A) =
z0 δ+θ
k=0
(1 + Sk + Tk ).
160
F. Aguiló-Gost, P.A. García-Sánchez / Electronic Notes in Discrete Mathematics 34 (2009) 157–161
Set F2 (m; A) = {(x, y) ∈ N2 | (x, y, z) ∈ F (m; A)}. The elements of F (m; A) are found by looking for the coordinates (x, y) ∈ F2 (m; A) from the starting point (x0 , y0 ) (part of the basic factorization of m related to H) and following the plane tessellation by H through the vectors u, v and e = u + v. The tree-like approach of F2 (87; A), A = {5, 7, 11}, is depicted in the figure; here we have H = L(5, 3, 2, 2), u = (5, −2), v = (−2, 3), e = (3, 1), δ = θ = 1, 0 ≤ k ≤ 3, {Sk }3k=0 = {0, 0, 1, 1} and {Tk }3k=0 = {1, 2, 3, 1}. Hence F (87; A) = {(2, 0, 7), (0, 3, 6), (5, 1, 5), (3, 4, 4), (1, 7, 3), (8, 2, 3), (13, 0, 2), (6, 5, 2), (4, 8, 1), (2, 11, 0), (11, 3, 1), (16, 1, 0), (9, 6, 0)}. 98 103 108 113 118 123 128 133 138 143 148 153 158 163 168 173 178 183 188 193 91 96 101 106 111 116 121 126 131 136 141 146 151 156 161 166 171 176 181 186 84 89 94 99 104 109 114 119 124 129 134 139 144 149 154 159 164 169 174 179 77 82 87 92 97 102 107 112 117 122 127 132 137 142 147 152 157 162 167 172 70 75 80 85 90 95 100 105 110 115 120 125 130 135 140 145 150 155 160 165 63 68 73 78 83 88 93 98 103 108 113 118 123 128 133 138 143 148 153 158 56 61 66 71 76 81 86 91 96 101 106 111 116 121 126 131 136 141 146 151 49 54 59 64 69 74 79 84 89 94 99 104 109 114 119 124 129 134 139 144 42 47 52 57 62 67 72 77 82 87 92 97 102 107 112 117 122 127 132 137 35 40 45 50 55 60 65 70 75 80 85 90 95 100 105 110 115 120 125 130 28 33 38 43 48 53 58 63 68 73 78 83 88 93 98 103 108 113 118 123 21 26 31 36 41 46 51 56 61 66 71 76 81 86 91 96 101 106 111 116 14 19 24 29 34 39 44 49 54 59 64 69 74 79 84 89 94 99 104 109 7 12 17 22 27 32 37 42 47 52 57 62 67 72 77 82 87 92 97 102 0
3
5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95
Catenary Degree
From now on let us assume A = {a, b, N }, 0 < a < b < N, gcd(a, b, N ) = 1, G(N ; a, b; a, b) has the related minimum distance diagram H = L(l, h, w, y), δ and θ are given in Definition 2.2. Set α = (l, −y, −δ), β = (−w, h, −θ) and γ = α + β = (l − w, h − y, −(δ + θ)). Lemma 3.1 α = l, β = max{h, w + θ}, γ = l − w + h − y. From the tree like shape described in Theorem 2.3 for the set F (m; A), with m ∈ S(A), one easily deduces the following consequence. Corollary 3.2 c(A) ≤ max{l, w + θ, h, l − w + h − y}. The idea of the proof of the following theorem is based on the concept of R-classes. See [2] for more details. Theorem 3.3 •
If w = 0, then c(A) = max{l, h}.
F. Aguiló-Gost, P.A. García-Sánchez / Electronic Notes in Discrete Mathematics 34 (2009) 157–161 •
If y = 0, then
•
If wy = 0, then
161
⎧ ⎨ max{w, l + l (h − w)} if θ = 0, w c(A) = ⎩ max{l, w + θ, h} otherwise.
c(A) =
⎧ ⎪ ⎪ max{l, l − w + h − y} ⎪ ⎨
if δ = 0,
max{w, l − y + wl (h − w)} if θ = 0, ⎪ ⎪ ⎪ ⎩ max{l, w + θ, h, l − w + h − y} otherwise.
References [1] Aguil´ o-Gost, F. and J. Barguilla, Computing coordinates inside an L-shaped tile, VI Jornadas de Matem´atica Discreta y Algor´ıtmica, Ed. Josep Conde et al., Edicions i Publicacions de la UdL, Universitat de Lleida, Spain (2008), 35–41. [2] Chapman, S. T., P. A. Garc´ıa-S´ anchez, D. Llena, V. Ponomarenko, and J. C. Rosales, The catenary and tame degree in finitely generated cancellative monoids, Manuscripta Math. 120 (2006), 253–264. [3] Chapman, S. T., P. A. Garc´ıa-S´ anchez, D. Llena, The catenary and tame degree of a numerical semigroup, Forum Math. 21 (2009), 117–129. [4] Cheng Y. and F. Hwang, Diameters of Weighted Double-Loop Networks, J. Algorithms 9 (1988), 401–410. [5] Delgado, M., P. A. Garc´ıa-S´ anchez and J. Morais, “numericalsgps”: a GAP [9] package on numerical semigroups. (http://www.gap-system.org/Packages/numericalsgps.html). [6] Fiol, M.A., J.L.A. Yebra, I. Alegre and M. Valero, A discrete optimization problem in local networks and data alignment, IEEE Trans. Comput. C-36 (1987), 702–713. [7] Ram´ırez Alfons´ın, J.L., “The Diophantine Frobenius Problem”. Oxford Univ. Press (2005) Oxford. ISBN 0-19-856820-7 978-0-19-856820-9. ¨ [8] R¨ odseth, Oystein J., On a linear Diophantine problem of Frobenius, J. Reine Angew. Math. 301 (1978) 171–178. [9] The GAP Group. GAP – Groups, Algorithms, and Programming, Version 4.4, 2004. (http://www.gap-system.org).