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 .

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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 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 ′).

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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, 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 ′).

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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.

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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.

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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.

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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).

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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).

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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.

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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

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 .

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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 + 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 .

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The end!