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 multiple
branches
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INTRODUCTION AND MOTIVATION
What is the normalization of a curve with multiple
branches
←
1/24
INTRODUCTION AND MOTIVATION
What is the normalization of a curve with multiple
branches
←
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]]/(x 3 − yz, y 3 − z 2 ) = C[[t 5 , t 6 , t 9 ]].
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]]/(x 3 − yz, y 3 − z 2 ) = C[[t 5 , t 6 , t 9 ]].
Then R̄ = C[[t]] is the normalization.
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]]/(x 3 − yz, y 3 − z 2 ) = C[[t 5 , t 6 , t 9 ]].
Then R̄ = C[[t]] is the normalization. The value semigroup of this
curve is
S =< 5, 6, 9 >= {0, 5, 6, 9, 10, 11, 12, 14 . . . }
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]]/(x 3 − yz, y 3 − z 2 ) = C[[t 5 , t 6 , t 9 ]].
Then R̄ = C[[t]] is the normalization. The value semigroup of this
curve is
S =< 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 13 γ
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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 )
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
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
a parametrization
γ
x 7→ (t1 , t25 ) + O(t13 , t211 )
y 7→ (0, −t23 )
and then the semigroup is
S =< (1, 5), (2, 9), (1, 3) + Ne1 , (3, 15) + Ne2 >
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INTRODUCTION AND MOTIVATION
Algebraic setting
a. k : algebraically closed field of arbitrary characteristic;
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, reduced
1-dimensional k -algebra;
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INTRODUCTION AND MOTIVATION
Algebraic setting
a. k : algebraically closed field of arbitrary characteristic;
b. (R, m): algebraic curve, i.e. a local, complete, Noetherian, reduced
1-dimensional k -algebra;
c. {p1 , . . . , pn } = Ass(R);
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, reduced
1-dimensional k -algebra;
c. {p1 , . . . , pn } = Ass(R);
d. R/pi , i ∈ {1, . . . , n}: branches of the curve;
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, reduced
1-dimensional k -algebra;
c. {p1 , . . . , pn } = Ass(R);
d. R/pi , i ∈ {1, . . . , n}: branches of the curve;
ϕ
e. R/pi ∼
= k [[ti ]]: discrete valuation ring (DVR);
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, reduced
1-dimensional k -algebra;
c. {p1 , . . . , pn } = Ass(R);
d. R/pi , i ∈ {1, . . . , n}: branches of the curve;
ϕ
e. R/pi ∼
= k [[ti ]]: discrete valuation ring (DVR);
f. νi valuation on the i-th branch: z ∈ R/pi 7→ ord(ϕ(z)) (see e);
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, reduced
1-dimensional k -algebra;
c. {p1 , . . . , pn } = Ass(R);
d. R/pi , i ∈ {1, . . . , n}: branches of the curve;
ϕ
e. R/pi ∼
= 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 ))
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(R reg ) ⊆ Nn .
where R reg = {x ∈ R | x non zero-divisor}.
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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(R reg ) ⊆ Nn .
where R reg = {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.
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INTRODUCTION AND MOTIVATION
Proprieties of value semigroups
Value semigroups are important because properties of the ring R can
be detected through the semigroup ΓR (e.g. Gorensteinness).
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INTRODUCTION AND MOTIVATION
Proprieties of value semigroups
Value semigroups are important because properties of the ring R can
be detected through the semigroup ΓR (e.g. Gorensteinness).
If the field k is "big enough", value semigroups of ideals always satisfy
the following:
ΓE + ΓR ⊆ ΓE
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INTRODUCTION AND MOTIVATION
Proprieties of value semigroups
Value semigroups are important because properties of the ring R can
be detected through the semigroup ΓR (e.g. Gorensteinness).
If the field k is "big enough", value semigroups of ideals always satisfy
the following:
ΓE + ΓR ⊆ ΓE
∃ α ∈ Zn such that α + Nn ⊆ ΓE .
7/24
INTRODUCTION AND MOTIVATION
Proprieties of value semigroups
Value semigroups are important because properties of the ring R can
be detected through the semigroup ΓR (e.g. Gorensteinness).
If the field k is "big enough", value semigroups of ideals always satisfy
the following:
ΓE + ΓR ⊆ ΓE
∃ α ∈ Zn such that α + Nn ⊆ ΓE .
min(α, β) ∈ ΓE ∀ α, β ∈ ΓE .
7/24
INTRODUCTION AND MOTIVATION
Proprieties of value semigroups
Value semigroups are important because properties of the ring R can
be detected through the semigroup ΓR (e.g. Gorensteinness).
If the field k is "big enough", value semigroups of ideals always satisfy
the 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 and
εi ≥ min(αi , βi ) for i 6= j (with equality if αi 6= βi ).
where E is a regular (fractional) ideal of R.
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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 for
some α ∈ Zn .
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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 for
some α ∈ Zn .
A semigroup ideal can satisfy the following properties:
(E0) ∃ α ∈ Zn such that α + Nn ⊆ E.
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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 for
some 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.
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 for
some 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 that
εj > 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
9/24
GOOD SEMIGROUPS
Properties of semigroup ideals
E satisfies (E0)
α
9/24
GOOD SEMIGROUPS
Properties of semigroup ideals
E satisfies (E0)
α
α
β
9/24
GOOD SEMIGROUPS
Properties of semigroup ideals
E satisfies (E0)
E satisfies (E1)
α
α
min{α, β}
β
9/24
GOOD SEMIGROUPS
Properties of semigroup ideals
E satisfies (E0)
E satisfies (E1)
α
α
min{α, β}
β
β
α
9/24
GOOD SEMIGROUPS
Properties of semigroup ideals
E satisfies (E0)
E satisfies (E1)
α
α
min{α, β}
β
E satisfies (E2)
β
α
ε
9/24
GOOD SEMIGROUPS
Good semigroups and good semigroup ideals
Definition (Good semigroup)
S is good if it satisfies (E0), (E1) and (E2).
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).
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 ideal
because any semigroup ideal of a good semigroup satisfies (E0).
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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 ideal
because any semigroup ideal of a good semigroup satisfies (E0).
Hence, if R is an algebraic curve
S := ΓR is a good semigroup
The 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)
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 it
is interesting to study good semigroups by themselves.
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GOOD SEMIGROUPS
Small elements
From now on, S will always be a good semigroup, and E a good
semigroup ideal of S.
E has a minimum
µE := min E
12/24
GOOD SEMIGROUPS
Small elements
From now on, S will always be a good semigroup, and E a good
semigroup ideal of S.
E has a minimum
µE := min E
and a conductor
γ E := µCE = min(α ∈ Zn | α + Nn ⊆ E)
where CE = γ E + Nn is the conductor ideal of E.
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GOOD SEMIGROUPS
Small elements
From now on, S will always be a good semigroup, and E a good
semigroup ideal of S.
E has a minimum
µE := min E
and a conductor
γ E := µCE = min(α ∈ Zn | α + Nn ⊆ E)
where CE = γ E + Nn is the conductor ideal of E.
Definition (Small elements)
Small(E) = {α ∈ E | α ≤ γ E } = min(γ E , E).
In particular, if E = S, we denote γ = γ S and
Small(S) = {α ∈ S | α ≤ γ} = min(γ, S).
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GOOD SEMIGROUPS
Small elements
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, γ E ) ∈ Small(E).
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GOOD SEMIGROUPS
Small elements determine the whole semigroup (ideal)
It is well-known that for good semigroup ideals:
Proposition
α ∈ E ⇐⇒ min(α, γ E ) ∈ Small(E).
Corollary
Let S and S 0 be two good semigroups. Then
S = S 0 ⇐⇒ Small(S) = Small(S 0 ).
Corollary
0
Let E and E 0 be two good semigroup ideals of S with γ E = γ E . Then
E = E 0 ⇐⇒ Small(E) = Small(E 0 ).
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GOOD GENERATING SYSTEMS FOR SEMIGROUPS
Good generating systems
Let G ⊂ Nn , and let
hGi = {g1 + · · · + gm | m ∈ N, g1 , . . . gn ∈ G}.
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GOOD GENERATING SYSTEMS FOR SEMIGROUPS
Good generating systems
Let G ⊂ Nn , and let
hGi = {g1 + · · · + gm | m ∈ N, g1 , . . . gn ∈ G}.
Set [G] to be the smallest submonoid of Nn containing G which is
closed under addition and minimums. Then
[G] = {min(g1 , . . . , gn ) | gi ∈ hGi}.
Denote
[G]γ := min(γ, [G]).
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GOOD GENERATING SYSTEMS FOR SEMIGROUPS
Good generating systems
Let G ⊂ Nn , and let
hGi = {g1 + · · · + gm | m ∈ N, g1 , . . . gn ∈ G}.
Set [G] to be the smallest submonoid of Nn containing G which is
closed under addition and minimums. Then
[G] = {min(g1 , . . . , gn ) | gi ∈ hGi}.
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.
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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) :=
S
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) :=
S
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) :=
S
i∈I
∆i (α).
15/24
GOOD GENERATING SYSTEMS FOR SEMIGROUPS
What do the ∆s actually mean
S
α
16/24
GOOD GENERATING SYSTEMS FOR SEMIGROUPS
What do the ∆s actually mean
S
α
∆1 (α)
α
16/24
GOOD GENERATING SYSTEMS FOR SEMIGROUPS
What do the ∆s actually mean
S
α
∆1 (α)
α
∆(α)
α
16/24
GOOD GENERATING SYSTEMS FOR SEMIGROUPS
What do the ∆s actually mean
S
α
∆(α)
α
∆1 (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 zero
components. From now on, we assume S to be local.
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 zero
components. From now on, we assume S to be local.
We want to be able to "take away" the unnecessary elements of a
good generating system, in order to get a minimal one. Hence the
following lemmas:
Lemma
G GGS for S, α ∈ G.
α ∈ [G \ {α}]γ =⇒ G \ {α} GGS for S.
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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 zero
components. From now on, we assume S to be local.
We want to be able to "take away" the unnecessary elements of a
good generating system, in order to get a minimal one. Hence the
following lemmas:
Lemma
G 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) ∩ hG \ {α}i =⇒ G \ {α} GGS for S.
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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


∆(α) ∩ hG \ {α}i = ∅ if Jα = ∅
G is a minimal GGS ⇐⇒ or otherwise


∆i (α) ∩ hG \ {α}i = ∅ for some i 6∈ Jα
for all α ∈ G.
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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


∆(α) ∩ hG \ {α}i = ∅ if Jα = ∅
G is a minimal GGS ⇐⇒ or otherwise


∆i (α) ∩ hG \ {α}i = ∅ for some i 6∈ Jα
for all α ∈ G.
Theorem (D’Anna, Garcia-Sanchez, Micale, T.)
S has a unique minimal GGS.
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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}.
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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 + S
which is closed under minimums. Then
{G} = {min(g1 , . . . , gn ) | gi ∈ G + S}.
Denote
{G}γ E := min(γ E , {G}).
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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 + S
which is closed under minimums. Then
{G} = {min(g1 , . . . , gn ) | gi ∈ G + S}.
Denote
{G}γ E := min(γ E , {G}).
Definition (good generating system)
G is a good generating system for E if {G}γ E = Small(E).
G is minimal if no proper subset of G is a good generating system of E.
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GOOD GENERATING SYSTEMS FOR SEMIGROUP IDEALS
Reducing a generic GGS to a minimal one
As before, we need to assume S local.
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 good
generating system, in order to get a minimal one.
We have an analogous of the provious lemm:
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 good
generating 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 γ E 6∈ ∆(α).
∃ β ∈ ∆(α) ∩ (G \ {α} + S) =⇒ G \ {α} GGS for E.
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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 γ E ∈ ∆Jα (α). Then


∆(α) ∩ (G \ {α} + S) = ∅ if Jα = ∅
G is a minimal GGS ⇐⇒ or otherwise


∆i (α) ∩ (G \ {α} + S) = ∅ for some i 6∈ Jα
for all α ∈ G.
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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 γ E ∈ ∆Jα (α). Then


∆(α) ∩ (G \ {α} + S) = ∅ if Jα = ∅
G is a minimal GGS ⇐⇒ 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:
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.
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
Remark
Every good semigroup is a direct product of good local semigroups.
24/24
EXAMPLES
The local assumption is necessary
Remark
Every good semigroup is a direct product of good local semigroups.
However, minimal GGSs of non-local semigroups do not need to be
unique:
24/24
EXAMPLES
The local assumption is necessary
Remark
Every good semigroup is a direct product of good local semigroups.
However, minimal GGSs of non-local semigroups do not need to be
unique:
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 minimal
GGSs for S × T .
24/24
The end!