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Good semigroups of Nn
PhD Seminar
Laura Tozzo
Universitá di Genova
Technische Universität Kaiserslautern
Genova, 06 April 2017
joint work with M. D’anna, P. Garcia-Sanchez and V. Micale
1 INTRODUCTION AND MOTIVATION
2 GOOD SEMIGROUPS
3 GOOD GENERATING SYSTEMS FOR SEMIGROUPS
4 GOOD GENERATING SYSTEMS FOR SEMIGROUP IDEALS
5 EXAMPLES
INTRODUCTION AND MOTIVATION
What is the normalization of a curve with multiplebranches
←
1/24
INTRODUCTION AND MOTIVATION
What is the normalization of a curve with multiplebranches
←
1/24
INTRODUCTION AND MOTIVATION
What is the normalization of a curve with multiplebranches
←
2/24
INTRODUCTION AND MOTIVATION
What is a value semigroup
Now take an irreducible curve (i.e. with only one branch):
R = C[[x , y , z]]/(x3 − yz, y3 − z2) = C[[t5, t6, t9]].
Then R̄ = C[[t ]] is the normalization. The value semigroup of this
curve isS =< 5,6,9 >= {0,5,6,9,10,11,12,14 . . . }
Namely, a subset of N given by the points
0 1 2 3 4 5 6 7 8 9 10 11 12 130 γ
3/24
INTRODUCTION AND MOTIVATION
What is a value semigroup
Now take an irreducible curve (i.e. with only one branch):
R = C[[x , y , z]]/(x3 − yz, y3 − z2) = C[[t5, t6, t9]].
Then R̄ = C[[t ]] is the normalization.
The value semigroup of this
curve isS =< 5,6,9 >= {0,5,6,9,10,11,12,14 . . . }
Namely, a subset of N given by the points
0 1 2 3 4 5 6 7 8 9 10 11 12 130 γ
3/24
INTRODUCTION AND MOTIVATION
What is a value semigroup
Now take an irreducible curve (i.e. with only one branch):
R = C[[x , y , z]]/(x3 − yz, y3 − z2) = C[[t5, t6, t9]].
Then R̄ = C[[t ]] is the normalization. The value semigroup of this
curve isS =< 5,6,9 >= {0,5,6,9,10,11,12,14 . . . }
Namely, a subset of N given by the points
0 1 2 3 4 5 6 7 8 9 10 11 12 130 γ
3/24
INTRODUCTION AND MOTIVATION
What is a value semigroup
Now take an irreducible curve (i.e. with only one branch):
R = C[[x , y , z]]/(x3 − yz, y3 − z2) = C[[t5, t6, t9]].
Then R̄ = C[[t ]] is the normalization. The value semigroup of this
curve isS =< 5,6,9 >= {0,5,6,9,10,11,12,14 . . . }
Namely, a subset of N given by the points
0 1 2 3 4 5 6 7 8 9 10 11 12 130 γ
3/24
INTRODUCTION AND MOTIVATION
What is a value semigroup - more in general
If the curve has more than one branch, like in this case:
R = C[[X ,Y ]]/Y (X 3 + Y 5)
Then the normalization is R̄ = C[[t1]]× C[[t2]]. We need
γ
a parametrization
x 7→ (t1, t52 ) +O(t3
1 , t112 )
y 7→ (0,−t32 )
and then the semigroup is
S =< (1,5), (2,9), (1,3) + Ne1, (3,15) + Ne2 >
4/24
INTRODUCTION AND MOTIVATION
What is a value semigroup - more in general
If the curve has more than one branch, like in this case:
R = C[[X ,Y ]]/Y (X 3 + Y 5)
Then the normalization is R̄ = C[[t1]]× C[[t2]]. We need
γ
a parametrization
x 7→ (t1, t52 ) +O(t3
1 , t112 )
y 7→ (0,−t32 )
and then the semigroup is
S =< (1,5), (2,9), (1,3) + Ne1, (3,15) + Ne2 >
4/24
INTRODUCTION AND MOTIVATION
What is a value semigroup - more in general
If the curve has more than one branch, like in this case:
R = C[[X ,Y ]]/Y (X 3 + Y 5)
Then the normalization is R̄ = C[[t1]]× C[[t2]]. We need
γ
a parametrization
x 7→ (t1, t52 ) +O(t3
1 , t112 )
y 7→ (0,−t32 )
and then the semigroup is
S =< (1,5), (2,9), (1,3) + Ne1, (3,15) + Ne2 >
4/24
INTRODUCTION AND MOTIVATION
Algebraic setting
a. k : algebraically closed field of arbitrary characteristic;
b. (R,m): algebraic curve, i.e. a local, complete, Noetherian, reduced1-dimensional k -algebra;
c. {p1, . . . , pn} = Ass(R);
d. R/pi , i ∈ {1, . . . ,n}: branches of the curve;
e. R/piϕ∼= k [[ti ]]: discrete valuation ring (DVR);
f. νi valuation on the i-th branch: z ∈ R/pi 7→ ord(ϕ(z)) (see e);
g. ν : R → R ∼= k [[t1]]× · · · × k [[ts]]→ Nn.
5/24
INTRODUCTION AND MOTIVATION
Algebraic setting
a. k : algebraically closed field of arbitrary characteristic;
b. (R,m): algebraic curve, i.e. a local, complete, Noetherian, reduced1-dimensional k -algebra;
c. {p1, . . . , pn} = Ass(R);
d. R/pi , i ∈ {1, . . . ,n}: branches of the curve;
e. R/piϕ∼= k [[ti ]]: discrete valuation ring (DVR);
f. νi valuation on the i-th branch: z ∈ R/pi 7→ ord(ϕ(z)) (see e);
g. ν : R → R ∼= k [[t1]]× · · · × k [[ts]]→ Nn.
5/24
INTRODUCTION AND MOTIVATION
Algebraic setting
a. k : algebraically closed field of arbitrary characteristic;
b. (R,m): algebraic curve, i.e. a local, complete, Noetherian, reduced1-dimensional k -algebra;
c. {p1, . . . , pn} = Ass(R);
d. R/pi , i ∈ {1, . . . ,n}: branches of the curve;
e. R/piϕ∼= k [[ti ]]: discrete valuation ring (DVR);
f. νi valuation on the i-th branch: z ∈ R/pi 7→ ord(ϕ(z)) (see e);
g. ν : R → R ∼= k [[t1]]× · · · × k [[ts]]→ Nn.
5/24
INTRODUCTION AND MOTIVATION
Algebraic setting
a. k : algebraically closed field of arbitrary characteristic;
b. (R,m): algebraic curve, i.e. a local, complete, Noetherian, reduced1-dimensional k -algebra;
c. {p1, . . . , pn} = Ass(R);
d. R/pi , i ∈ {1, . . . ,n}: branches of the curve;
e. R/piϕ∼= k [[ti ]]: discrete valuation ring (DVR);
f. νi valuation on the i-th branch: z ∈ R/pi 7→ ord(ϕ(z)) (see e);
g. ν : R → R ∼= k [[t1]]× · · · × k [[ts]]→ Nn.
5/24
INTRODUCTION AND MOTIVATION
Algebraic setting
a. k : algebraically closed field of arbitrary characteristic;
b. (R,m): algebraic curve, i.e. a local, complete, Noetherian, reduced1-dimensional k -algebra;
c. {p1, . . . , pn} = Ass(R);
d. R/pi , i ∈ {1, . . . ,n}: branches of the curve;
e. R/piϕ∼= k [[ti ]]: discrete valuation ring (DVR);
f. νi valuation on the i-th branch: z ∈ R/pi 7→ ord(ϕ(z)) (see e);
g. ν : R → R ∼= k [[t1]]× · · · × k [[ts]]→ Nn.
5/24
INTRODUCTION AND MOTIVATION
Algebraic setting
a. k : algebraically closed field of arbitrary characteristic;
b. (R,m): algebraic curve, i.e. a local, complete, Noetherian, reduced1-dimensional k -algebra;
c. {p1, . . . , pn} = Ass(R);
d. R/pi , i ∈ {1, . . . ,n}: branches of the curve;
e. R/piϕ∼= k [[ti ]]: discrete valuation ring (DVR);
f. νi valuation on the i-th branch: z ∈ R/pi 7→ ord(ϕ(z)) (see e);
g. ν : R → R ∼= k [[t1]]× · · · × k [[ts]]→ Nn.
5/24
INTRODUCTION AND MOTIVATION
Algebraic setting
a. k : algebraically closed field of arbitrary characteristic;
b. (R,m): algebraic curve, i.e. a local, complete, Noetherian, reduced1-dimensional k -algebra;
c. {p1, . . . , pn} = Ass(R);
d. R/pi , i ∈ {1, . . . ,n}: branches of the curve;
e. R/piϕ∼= k [[ti ]]: discrete valuation ring (DVR);
f. νi valuation on the i-th branch: z ∈ R/pi 7→ ord(ϕ(z)) (see e);
g. ν : R → R ∼= k [[t1]]× · · · × k [[ts]]→ Nn.
5/24
INTRODUCTION AND MOTIVATION
Definition of value semigroup
Let R be the ring of an algebraic curve.
ν : R → Nn
x 7→ (ord(x(t1)), . . . ,ord(x(tn))
Definition (Value semigroup of an algebraic curve)
ΓR = ν(Rreg) ⊆ Nn.
where Rreg = {x ∈ R | x non zero-divisor}.
A (fractional) ideal of R is regular if it contains a non zero-divisor.
Definition (Value semigroup of an ideal)
ΓE := ν(E reg) ⊆ Zn ∀ E regular (fractional) ideal of R.
6/24
INTRODUCTION AND MOTIVATION
Definition of value semigroup
Let R be the ring of an algebraic curve.
ν : R → Nn
x 7→ (ord(x(t1)), . . . ,ord(x(tn))
Definition (Value semigroup of an algebraic curve)
ΓR = ν(Rreg) ⊆ Nn.
where Rreg = {x ∈ R | x non zero-divisor}.
A (fractional) ideal of R is regular if it contains a non zero-divisor.
Definition (Value semigroup of an ideal)
ΓE := ν(E reg) ⊆ Zn ∀ E regular (fractional) ideal of R.
6/24
INTRODUCTION AND MOTIVATION
Definition of value semigroup
Let R be the ring of an algebraic curve.
ν : R → Nn
x 7→ (ord(x(t1)), . . . ,ord(x(tn))
Definition (Value semigroup of an algebraic curve)
ΓR = ν(Rreg) ⊆ Nn.
where Rreg = {x ∈ R | x non zero-divisor}.
A (fractional) ideal of R is regular if it contains a non zero-divisor.
Definition (Value semigroup of an ideal)
ΓE := ν(E reg) ⊆ Zn ∀ E regular (fractional) ideal of R.
6/24
INTRODUCTION AND MOTIVATION
Proprieties of value semigroups
Value semigroups are important because properties of the ring R canbe detected through the semigroup ΓR (e.g. Gorensteinness).
If the field k is "big enough", value semigroups of ideals always satisfythe following:
ΓE + ΓR ⊆ ΓE
∃ α ∈ Zn such that α + Nn ⊆ ΓE .
min(α, β) ∈ ΓE ∀ α, β ∈ ΓE .
∀ α, β ∈ ΓE such that αj = βj , ∃ ε ∈ ΓE s. t. ε > αj = βj andεi ≥ min(αi , βi) for i 6= j (with equality if αi 6= βi ).
where E is a regular (fractional) ideal of R.
7/24
INTRODUCTION AND MOTIVATION
Proprieties of value semigroups
Value semigroups are important because properties of the ring R canbe detected through the semigroup ΓR (e.g. Gorensteinness).
If the field k is "big enough", value semigroups of ideals always satisfythe following:
ΓE + ΓR ⊆ ΓE
∃ α ∈ Zn such that α + Nn ⊆ ΓE .
min(α, β) ∈ ΓE ∀ α, β ∈ ΓE .
∀ α, β ∈ ΓE such that αj = βj , ∃ ε ∈ ΓE s. t. ε > αj = βj andεi ≥ min(αi , βi) for i 6= j (with equality if αi 6= βi ).
where E is a regular (fractional) ideal of R.
7/24
INTRODUCTION AND MOTIVATION
Proprieties of value semigroups
Value semigroups are important because properties of the ring R canbe detected through the semigroup ΓR (e.g. Gorensteinness).
If the field k is "big enough", value semigroups of ideals always satisfythe following:
ΓE + ΓR ⊆ ΓE
∃ α ∈ Zn such that α + Nn ⊆ ΓE .
min(α, β) ∈ ΓE ∀ α, β ∈ ΓE .
∀ α, β ∈ ΓE such that αj = βj , ∃ ε ∈ ΓE s. t. ε > αj = βj andεi ≥ min(αi , βi) for i 6= j (with equality if αi 6= βi ).
where E is a regular (fractional) ideal of R.
7/24
INTRODUCTION AND MOTIVATION
Proprieties of value semigroups
Value semigroups are important because properties of the ring R canbe detected through the semigroup ΓR (e.g. Gorensteinness).
If the field k is "big enough", value semigroups of ideals always satisfythe following:
ΓE + ΓR ⊆ ΓE
∃ α ∈ Zn such that α + Nn ⊆ ΓE .
min(α, β) ∈ ΓE ∀ α, β ∈ ΓE .
∀ α, β ∈ ΓE such that αj = βj , ∃ ε ∈ ΓE s. t. ε > αj = βj andεi ≥ min(αi , βi) for i 6= j (with equality if αi 6= βi ).
where E is a regular (fractional) ideal of R.
7/24
INTRODUCTION AND MOTIVATION
Proprieties of value semigroups
Value semigroups are important because properties of the ring R canbe detected through the semigroup ΓR (e.g. Gorensteinness).
If the field k is "big enough", value semigroups of ideals always satisfythe following:
ΓE + ΓR ⊆ ΓE
∃ α ∈ Zn such that α + Nn ⊆ ΓE .
min(α, β) ∈ ΓE ∀ α, β ∈ ΓE .
∀ α, β ∈ ΓE such that αj = βj , ∃ ε ∈ ΓE s. t. ε > αj = βj andεi ≥ min(αi , βi) for i 6= j (with equality if αi 6= βi ).
where E is a regular (fractional) ideal of R.
7/24
GOOD SEMIGROUPS
Semigroups
A semigroup S ⊆ Nn is a set of integers containing 0 and closed w.r.t.the sum.
Definition (Semigroup ideal)
An ideal of S is an E ⊆ Zn such that E + S ⊂ E and α + E ⊆ Nn forsome α ∈ Zn.
A semigroup ideal can satisfy the following properties:
(E0) ∃ α ∈ Zn such that α + Nn ⊆ E .
(E1) If α, β ∈ E , then min(α, β) := (min(α1, β1), . . . ,min(αn, βn)) ∈ E .
(E2) ∀ α, β ∈ E and ∀ j ∈ {1, . . . ,n} such that αj = βj , ∃ ε ∈ E such thatεj > αj = βj and εi ≥ min(αi , βi) for all i 6= j , with equality if αi 6= βi .
8/24
GOOD SEMIGROUPS
Semigroups
A semigroup S ⊆ Nn is a set of integers containing 0 and closed w.r.t.the sum.
Definition (Semigroup ideal)
An ideal of S is an E ⊆ Zn such that E + S ⊂ E and α + E ⊆ Nn forsome α ∈ Zn.
A semigroup ideal can satisfy the following properties:
(E0) ∃ α ∈ Zn such that α + Nn ⊆ E .
(E1) If α, β ∈ E , then min(α, β) := (min(α1, β1), . . . ,min(αn, βn)) ∈ E .
(E2) ∀ α, β ∈ E and ∀ j ∈ {1, . . . ,n} such that αj = βj , ∃ ε ∈ E such thatεj > αj = βj and εi ≥ min(αi , βi) for all i 6= j , with equality if αi 6= βi .
8/24
GOOD SEMIGROUPS
Semigroups
A semigroup S ⊆ Nn is a set of integers containing 0 and closed w.r.t.the sum.
Definition (Semigroup ideal)
An ideal of S is an E ⊆ Zn such that E + S ⊂ E and α + E ⊆ Nn forsome α ∈ Zn.
A semigroup ideal can satisfy the following properties:
(E0) ∃ α ∈ Zn such that α + Nn ⊆ E .
(E1) If α, β ∈ E , then min(α, β) := (min(α1, β1), . . . ,min(αn, βn)) ∈ E .
(E2) ∀ α, β ∈ E and ∀ j ∈ {1, . . . ,n} such that αj = βj , ∃ ε ∈ E such thatεj > αj = βj and εi ≥ min(αi , βi) for all i 6= j , with equality if αi 6= βi .
8/24
GOOD SEMIGROUPS
Semigroups
A semigroup S ⊆ Nn is a set of integers containing 0 and closed w.r.t.the sum.
Definition (Semigroup ideal)
An ideal of S is an E ⊆ Zn such that E + S ⊂ E and α + E ⊆ Nn forsome α ∈ Zn.
A semigroup ideal can satisfy the following properties:
(E0) ∃ α ∈ Zn such that α + Nn ⊆ E .
(E1) If α, β ∈ E , then min(α, β) := (min(α1, β1), . . . ,min(αn, βn)) ∈ E .
(E2) ∀ α, β ∈ E and ∀ j ∈ {1, . . . ,n} such that αj = βj , ∃ ε ∈ E such thatεj > αj = βj and εi ≥ min(αi , βi) for all i 6= j , with equality if αi 6= βi .
8/24
GOOD SEMIGROUPS
Properties of semigroup ideals
α
E satisfies (E0)
α
βmin{α, β}
E satisfies (E1)
α
β
ε
E satisfies (E2)
9/24
GOOD SEMIGROUPS
Properties of semigroup ideals
α
E satisfies (E0)
α
βmin{α, β}
E satisfies (E1)
α
β
ε
E satisfies (E2)
9/24
GOOD SEMIGROUPS
Properties of semigroup ideals
α
E satisfies (E0)
α
β
min{α, β}
E satisfies (E1)
α
β
ε
E satisfies (E2)
9/24
GOOD SEMIGROUPS
Properties of semigroup ideals
α
E satisfies (E0)
α
βmin{α, β}
E satisfies (E1)
α
β
ε
E satisfies (E2)
9/24
GOOD SEMIGROUPS
Properties of semigroup ideals
α
E satisfies (E0)
α
βmin{α, β}
E satisfies (E1)
α
β
ε
E satisfies (E2)
9/24
GOOD SEMIGROUPS
Properties of semigroup ideals
α
E satisfies (E0)
α
βmin{α, β}
E satisfies (E1)
α
β
ε
E satisfies (E2)
9/24
GOOD SEMIGROUPS
Good semigroups and good semigroup ideals
Definition (Good semigroup)
S is good if it satisfies (E0), (E1) and (E2).
Definition (Good semigroup ideal)
E is good if it satisfies (E1) and (E2).
We do not require condition (E0) in the definition of good idealbecause any semigroup ideal of a good semigroup satisfies (E0).
Hence, if R is an algebraic curve
S := ΓR is a good semigroupThe semigroup ideals of S of the type ΓE are good
10/24
GOOD SEMIGROUPS
Good semigroups and good semigroup ideals
Definition (Good semigroup)
S is good if it satisfies (E0), (E1) and (E2).
Definition (Good semigroup ideal)
E is good if it satisfies (E1) and (E2).
We do not require condition (E0) in the definition of good idealbecause any semigroup ideal of a good semigroup satisfies (E0).
Hence, if R is an algebraic curve
S := ΓR is a good semigroupThe semigroup ideals of S of the type ΓE are good
10/24
GOOD SEMIGROUPS
Good semigroups and good semigroup ideals
Definition (Good semigroup)
S is good if it satisfies (E0), (E1) and (E2).
Definition (Good semigroup ideal)
E is good if it satisfies (E1) and (E2).
We do not require condition (E0) in the definition of good idealbecause any semigroup ideal of a good semigroup satisfies (E0).
Hence, if R is an algebraic curve
S := ΓR is a good semigroupThe semigroup ideals of S of the type ΓE are good
10/24
GOOD SEMIGROUPS
Good semigroups and good semigroup ideals
Definition (Good semigroup)
S is good if it satisfies (E0), (E1) and (E2).
Definition (Good semigroup ideal)
E is good if it satisfies (E1) and (E2).
We do not require condition (E0) in the definition of good idealbecause any semigroup ideal of a good semigroup satisfies (E0).
Hence, if R is an algebraic curve
S := ΓR is a good semigroupThe semigroup ideals of S of the type ΓE are good
10/24
GOOD SEMIGROUPS
Remarks
It can happen that ΓEF ( ΓE + ΓF .
S E
F E + F
E + F does not satisfy (E2)
Not all good semigroups are value semigroups. For this reason itis interesting to study good semigroups by themselves.
11/24
GOOD SEMIGROUPS
Remarks
It can happen that ΓEF ( ΓE + ΓF .
S E
F E + F
E + F does not satisfy (E2)
Not all good semigroups are value semigroups. For this reason itis interesting to study good semigroups by themselves.
11/24
GOOD SEMIGROUPS
Small elements
From now on, S will always be a good semigroup, and E a goodsemigroup ideal of S.E has a minimum
µE := min E
and a conductor
γE := µCE = min(α ∈ Zn | α + Nn ⊆ E)
where CE = γE + Nn is the conductor ideal of E .
Definition (Small elements)
Small(E) = {α ∈ E | α ≤ γE} = min(γE ,E).
In particular, if E = S, we denote γ = γS and
Small(S) = {α ∈ S | α ≤ γ} = min(γ,S).
12/24
GOOD SEMIGROUPS
Small elements
From now on, S will always be a good semigroup, and E a goodsemigroup ideal of S.E has a minimum
µE := min E
and a conductor
γE := µCE = min(α ∈ Zn | α + Nn ⊆ E)
where CE = γE + Nn is the conductor ideal of E .
Definition (Small elements)
Small(E) = {α ∈ E | α ≤ γE} = min(γE ,E).
In particular, if E = S, we denote γ = γS and
Small(S) = {α ∈ S | α ≤ γ} = min(γ,S).
12/24
GOOD SEMIGROUPS
Small elements
From now on, S will always be a good semigroup, and E a goodsemigroup ideal of S.E has a minimum
µE := min E
and a conductor
γE := µCE = min(α ∈ Zn | α + Nn ⊆ E)
where CE = γE + Nn is the conductor ideal of E .
Definition (Small elements)
Small(E) = {α ∈ E | α ≤ γE} = min(γE ,E).
In particular, if E = S, we denote γ = γS and
Small(S) = {α ∈ S | α ≤ γ} = min(γ,S).
12/24
GOOD SEMIGROUPS
Small elements
Small(E)
12/24
GOOD SEMIGROUPS
Small elements
Small(E)
12/24
GOOD SEMIGROUPS
Small elements determine the whole semigroup (ideal)
It is well-known that for good semigroup ideals:
Proposition
α ∈ E ⇐⇒ min(α, γE ) ∈ Small(E).
Corollary
Let S and S′ be two good semigroups. Then
S = S′ ⇐⇒ Small(S) = Small(S′).
Corollary
Let E and E ′ be two good semigroup ideals of S with γE = γE ′. Then
E = E ′ ⇐⇒ Small(E) = Small(E ′).
13/24
GOOD SEMIGROUPS
Small elements determine the whole semigroup (ideal)
It is well-known that for good semigroup ideals:
Proposition
α ∈ E ⇐⇒ min(α, γE ) ∈ Small(E).
Corollary
Let S and S′ be two good semigroups. Then
S = S′ ⇐⇒ Small(S) = Small(S′).
Corollary
Let E and E ′ be two good semigroup ideals of S with γE = γE ′. Then
E = E ′ ⇐⇒ Small(E) = Small(E ′).
13/24
GOOD GENERATING SYSTEMS FOR SEMIGROUPS
Good generating systems
Let G ⊂ Nn, and let
〈G〉 = {g1 + · · ·+ gm | m ∈ N,g1, . . .gn ∈ G}.
Set [G] to be the smallest submonoid of Nn containing G which isclosed under addition and minimums. Then
[G] = {min(g1, . . . ,gn) | gi ∈ 〈G〉}.
Denote[G]γ := min(γ, [G]).
Definition (good generating system)
G is a good generating system for S if [G]γ = Small(S).
G is minimal if no proper subset of G is a good generating system of S.
14/24
GOOD GENERATING SYSTEMS FOR SEMIGROUPS
Good generating systems
Let G ⊂ Nn, and let
〈G〉 = {g1 + · · ·+ gm | m ∈ N,g1, . . .gn ∈ G}.
Set [G] to be the smallest submonoid of Nn containing G which isclosed under addition and minimums. Then
[G] = {min(g1, . . . ,gn) | gi ∈ 〈G〉}.
Denote[G]γ := min(γ, [G]).
Definition (good generating system)
G is a good generating system for S if [G]γ = Small(S).
G is minimal if no proper subset of G is a good generating system of S.
14/24
GOOD GENERATING SYSTEMS FOR SEMIGROUPS
Good generating systems
Let G ⊂ Nn, and let
〈G〉 = {g1 + · · ·+ gm | m ∈ N,g1, . . .gn ∈ G}.
Set [G] to be the smallest submonoid of Nn containing G which isclosed under addition and minimums. Then
[G] = {min(g1, . . . ,gn) | gi ∈ 〈G〉}.
Denote[G]γ := min(γ, [G]).
Definition (good generating system)
G is a good generating system for S if [G]γ = Small(S).
G is minimal if no proper subset of G is a good generating system of S.
14/24
GOOD GENERATING SYSTEMS FOR SEMIGROUPS
Some technical definitions
∆J(α) := {β ∈ Zn | αj = βj for j ∈ J and αi < βi for i ∈ I \ J}.
If J = {i} we denote ∆J = ∆i .
∆(α) :=⋃
i∈I ∆i(α).
∆J(α) := {β ∈ Zn | αj = βj for j ∈ J and αi ≤ βi for i ∈ I \ J}.
If J = {i} we denote ∆J = ∆i .
∆(α) :=⋃
i∈I ∆i(α).
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GOOD GENERATING SYSTEMS FOR SEMIGROUPS
Some technical definitions
∆J(α) := {β ∈ Zn | αj = βj for j ∈ J and αi < βi for i ∈ I \ J}.
If J = {i} we denote ∆J = ∆i .
∆(α) :=⋃
i∈I ∆i(α).
∆J(α) := {β ∈ Zn | αj = βj for j ∈ J and αi ≤ βi for i ∈ I \ J}.
If J = {i} we denote ∆J = ∆i .
∆(α) :=⋃
i∈I ∆i(α).
15/24
GOOD GENERATING SYSTEMS FOR SEMIGROUPS
What do the ∆s actually mean
S
α
α
∆1(α)
α
∆(α)
α
∆(α)
16/24
GOOD GENERATING SYSTEMS FOR SEMIGROUPS
What do the ∆s actually mean
S
α α
∆1(α)
α
∆(α)
α
∆(α)
16/24
GOOD GENERATING SYSTEMS FOR SEMIGROUPS
What do the ∆s actually mean
S
α α
∆1(α)
α
∆(α)
α
∆(α)
16/24
GOOD GENERATING SYSTEMS FOR SEMIGROUPS
What do the ∆s actually mean
S
α α
∆1(α)
α
∆(α)
α
∆(α)
16/24
GOOD GENERATING SYSTEMS FOR SEMIGROUPS
Reducing a generic GGS to a minimal one
A semigroup is local if the zero is the only element with zerocomponents. From now on, we assume S to be local.
We want to be able to "take away" the unnecessary elements of agood generating system, in order to get a minimal one. Hence thefollowing lemmas:
LemmaG GGS for S, α ∈ G.
α ∈ [G \ {α}]γ =⇒ G \ {α} GGS for S.
Lemma
G GGS for S, α ∈ G such that γ 6∈ ∆(α).
∃ β ∈ ∆(α) ∩ 〈G \ {α}〉 =⇒ G \ {α} GGS for S.
17/24
GOOD GENERATING SYSTEMS FOR SEMIGROUPS
Reducing a generic GGS to a minimal one
A semigroup is local if the zero is the only element with zerocomponents. From now on, we assume S to be local.
We want to be able to "take away" the unnecessary elements of agood generating system, in order to get a minimal one. Hence thefollowing lemmas:
LemmaG GGS for S, α ∈ G.
α ∈ [G \ {α}]γ =⇒ G \ {α} GGS for S.
Lemma
G GGS for S, α ∈ G such that γ 6∈ ∆(α).
∃ β ∈ ∆(α) ∩ 〈G \ {α}〉 =⇒ G \ {α} GGS for S.
17/24
GOOD GENERATING SYSTEMS FOR SEMIGROUPS
Reducing a generic GGS to a minimal one
A semigroup is local if the zero is the only element with zerocomponents. From now on, we assume S to be local.
We want to be able to "take away" the unnecessary elements of agood generating system, in order to get a minimal one. Hence thefollowing lemmas:
LemmaG GGS for S, α ∈ G.
α ∈ [G \ {α}]γ =⇒ G \ {α} GGS for S.
Lemma
G GGS for S, α ∈ G such that γ 6∈ ∆(α).
∃ β ∈ ∆(α) ∩ 〈G \ {α}〉 =⇒ G \ {α} GGS for S.
17/24
GOOD GENERATING SYSTEMS FOR SEMIGROUPS
Characterization and uniqueness of minimal GGSs
Theorem (D’Anna, Garcia-Sanchez, Micale, T.)
G GGS for S. For α ∈ G, let Jα be such that γ ∈ ∆Jα(α). Then
G is a minimal GGS ⇐⇒
∆(α) ∩ 〈G \ {α}〉 = ∅ if Jα = ∅or otherwise∆i(α) ∩ 〈G \ {α}〉 = ∅ for some i 6∈ Jα
for all α ∈ G.
Theorem (D’Anna, Garcia-Sanchez, Micale, T.)S has a unique minimal GGS.
18/24
GOOD GENERATING SYSTEMS FOR SEMIGROUPS
Characterization and uniqueness of minimal GGSs
Theorem (D’Anna, Garcia-Sanchez, Micale, T.)
G GGS for S. For α ∈ G, let Jα be such that γ ∈ ∆Jα(α). Then
G is a minimal GGS ⇐⇒
∆(α) ∩ 〈G \ {α}〉 = ∅ if Jα = ∅or otherwise∆i(α) ∩ 〈G \ {α}〉 = ∅ for some i 6∈ Jα
for all α ∈ G.
Theorem (D’Anna, Garcia-Sanchez, Micale, T.)S has a unique minimal GGS.
18/24
GOOD GENERATING SYSTEMS FOR SEMIGROUP IDEALS
Good generating system for semigroup ideals
Let G ⊂ Nn, and let
G + S = {g + s | g ∈ G, s ∈ S}.
Set {G} to be the smallest semigroup ideal of S containing G + Swhich is closed under minimums. Then
{G} = {min(g1, . . . ,gn) | gi ∈ G + S}.
Denote{G}γE := min(γE , {G}).
Definition (good generating system)
G is a good generating system for E if {G}γE = Small(E).
G is minimal if no proper subset of G is a good generating system of E .
19/24
GOOD GENERATING SYSTEMS FOR SEMIGROUP IDEALS
Good generating system for semigroup ideals
Let G ⊂ Nn, and let
G + S = {g + s | g ∈ G, s ∈ S}.
Set {G} to be the smallest semigroup ideal of S containing G + Swhich is closed under minimums. Then
{G} = {min(g1, . . . ,gn) | gi ∈ G + S}.
Denote{G}γE := min(γE , {G}).
Definition (good generating system)
G is a good generating system for E if {G}γE = Small(E).
G is minimal if no proper subset of G is a good generating system of E .
19/24
GOOD GENERATING SYSTEMS FOR SEMIGROUP IDEALS
Good generating system for semigroup ideals
Let G ⊂ Nn, and let
G + S = {g + s | g ∈ G, s ∈ S}.
Set {G} to be the smallest semigroup ideal of S containing G + Swhich is closed under minimums. Then
{G} = {min(g1, . . . ,gn) | gi ∈ G + S}.
Denote{G}γE := min(γE , {G}).
Definition (good generating system)
G is a good generating system for E if {G}γE = Small(E).
G is minimal if no proper subset of G is a good generating system of E .
19/24
GOOD GENERATING SYSTEMS FOR SEMIGROUP IDEALS
Reducing a generic GGS to a minimal one
As before, we need to assume S local.
Again, we want to "take away" the unnecessary elements of a goodgenerating system, in order to get a minimal one.
We have an analogous of the provious lemm:
Lemma
G GGS for E, α ∈ G such that γE 6∈ ∆(α).
∃ β ∈ ∆(α) ∩ (G \ {α}+ S) =⇒ G \ {α} GGS for E .
20/24
GOOD GENERATING SYSTEMS FOR SEMIGROUP IDEALS
Reducing a generic GGS to a minimal one
As before, we need to assume S local.
Again, we want to "take away" the unnecessary elements of a goodgenerating system, in order to get a minimal one.
We have an analogous of the provious lemm:
Lemma
G GGS for E, α ∈ G such that γE 6∈ ∆(α).
∃ β ∈ ∆(α) ∩ (G \ {α}+ S) =⇒ G \ {α} GGS for E .
20/24
GOOD GENERATING SYSTEMS FOR SEMIGROUP IDEALS
Reducing a generic GGS to a minimal one
As before, we need to assume S local.
Again, we want to "take away" the unnecessary elements of a goodgenerating system, in order to get a minimal one.
We have an analogous of the provious lemm:
Lemma
G GGS for E, α ∈ G such that γE 6∈ ∆(α).
∃ β ∈ ∆(α) ∩ (G \ {α}+ S) =⇒ G \ {α} GGS for E .
20/24
GOOD GENERATING SYSTEMS FOR SEMIGROUP IDEALS
Characterization and uniqueness of minimal GGSs
Theorem (D’Anna, Garcia-Sanchez, Micale, T.)
G GGS for E. For α ∈ G, let Jα be such that γE ∈ ∆Jα(α). Then
G is a minimal GGS ⇐⇒
∆(α) ∩ (G \ {α}+ S) = ∅ if Jα = ∅or otherwise∆i(α) ∩ (G \ {α}+ S) = ∅ for some i 6∈ Jα
for all α ∈ G.
Theorem (D’Anna, Garcia-Sanchez, Micale, T.)E has a unique minimal GGS.
21/24
GOOD GENERATING SYSTEMS FOR SEMIGROUP IDEALS
Characterization and uniqueness of minimal GGSs
Theorem (D’Anna, Garcia-Sanchez, Micale, T.)
G GGS for E. For α ∈ G, let Jα be such that γE ∈ ∆Jα(α). Then
G is a minimal GGS ⇐⇒
∆(α) ∩ (G \ {α}+ S) = ∅ if Jα = ∅or otherwise∆i(α) ∩ (G \ {α}+ S) = ∅ for some i 6∈ Jα
for all α ∈ G.
Theorem (D’Anna, Garcia-Sanchez, Micale, T.)E has a unique minimal GGS.
21/24
EXAMPLES
Not all sets can be GGSs...
Let G = {(2,2), (4,2)} and γ = (6,6). Then [G]γ looks like:
Condition (E2) does not hold: there should be an element in{(2,3), (2,4), (2,5), (2,6)} since (2,2) and (4,2) share a coordinate.
22/24
EXAMPLES
Not all sets can be GGSs...
Let G = {(2,2), (4,2)} and γ = (6,6). Then [G]γ looks like:
Condition (E2) does not hold: there should be an element in{(2,3), (2,4), (2,5), (2,6)} since (2,2) and (4,2) share a coordinate.
22/24
EXAMPLES
...even if they satisfy the characterization conditions
Even if G agrees with the conditions of the characterization theorem,the resulting semigroup might not be good.
Let G = {(3,4), (7,8)} and γ = (8,10). Then [G]γ is
23/24
EXAMPLES
...even if they satisfy the characterization conditions
Even if G agrees with the conditions of the characterization theorem,the resulting semigroup might not be good.
Let G = {(3,4), (7,8)} and γ = (8,10). Then [G]γ is
23/24
EXAMPLES
The local assumption is necessary
RemarkEvery good semigroup is a direct product of good local semigroups.
However, minimal GGSs of non-local semigroups do not need to beunique:gap> S:=NumericalSemigroup(3,5,7);<Numerical semigroup with 3 generators>gap> T:=NumericalSemigroup(2,5);<Modular numerical semigroup satisfying 5x mod 10 <= x >gap> W:=cartesianProduct(S,T);<Good semigroup>gap> SmallElementsOfGoodSemigroup(W);[ [ 0, 0 ], [ 0, 2 ], [ 0, 4 ], [ 3, 0 ], [ 3, 2 ],[ 3, 4 ], [ 5, 0 ], [ 5, 2 ], [ 5, 4 ] ]
Both {(0,4), (3,2), (5,0)} and {(0,4), (3,4), (5,0), (5,2)} are minimalGGSs for S × T .
24/24
EXAMPLES
The local assumption is necessary
RemarkEvery good semigroup is a direct product of good local semigroups.
However, minimal GGSs of non-local semigroups do not need to beunique:
gap> S:=NumericalSemigroup(3,5,7);<Numerical semigroup with 3 generators>gap> T:=NumericalSemigroup(2,5);<Modular numerical semigroup satisfying 5x mod 10 <= x >gap> W:=cartesianProduct(S,T);<Good semigroup>gap> SmallElementsOfGoodSemigroup(W);[ [ 0, 0 ], [ 0, 2 ], [ 0, 4 ], [ 3, 0 ], [ 3, 2 ],[ 3, 4 ], [ 5, 0 ], [ 5, 2 ], [ 5, 4 ] ]
Both {(0,4), (3,2), (5,0)} and {(0,4), (3,4), (5,0), (5,2)} are minimalGGSs for S × T .
24/24
EXAMPLES
The local assumption is necessary
RemarkEvery good semigroup is a direct product of good local semigroups.
However, minimal GGSs of non-local semigroups do not need to beunique:gap> S:=NumericalSemigroup(3,5,7);<Numerical semigroup with 3 generators>gap> T:=NumericalSemigroup(2,5);<Modular numerical semigroup satisfying 5x mod 10 <= x >gap> W:=cartesianProduct(S,T);<Good semigroup>gap> SmallElementsOfGoodSemigroup(W);[ [ 0, 0 ], [ 0, 2 ], [ 0, 4 ], [ 3, 0 ], [ 3, 2 ],[ 3, 4 ], [ 5, 0 ], [ 5, 2 ], [ 5, 4 ] ]
Both {(0,4), (3,2), (5,0)} and {(0,4), (3,4), (5,0), (5,2)} are minimalGGSs for S × T .
24/24
The end!