Some Aspects of Weak Ideal Topology on the Topos of Right Acts over - - PowerPoint PPT Presentation

Preliminaries Weak Ideal Topology Results References

Some Aspects of Weak Ideal Topology on the Topos of Right Acts over a Monoid

Ali Madanshekaf

Semnan University amadanshekaf@semnan.ac.ir

BLAST 2018 University of Denver, Colorado, USA August 6-10 2018

Ali Madanshekaf Semnan University

Preliminaries Weak Ideal Topology Results References

Outline

Overview of talk In this talk we

Ali Madanshekaf Semnan University

Preliminaries Weak Ideal Topology Results References

Outline

Overview of talk In this talk we introduce the (weak) ideal topology jI on the topos Act-S of right S-acts. Afterwards, we observe that some known categories of separated acts over a monoid are special cases of jI-separated categories.

Ali Madanshekaf Semnan University

Preliminaries Weak Ideal Topology Results References

Outline

Overview of talk In this talk we introduce the (weak) ideal topology jI on the topos Act-S of right S-acts. Afterwards, we observe that some known categories of separated acts over a monoid are special cases of jI-separated categories. give a simple description of the associated sheaf functor with respect to the ideal topology jI where I is a central band of S.

Ali Madanshekaf Semnan University

Preliminaries Weak Ideal Topology Results References Weak Topologies and Modal Closure Operators

Definition

Let S be a monoid with the identity 1. A (right) S-act is a set A equipped with an action µ : A × S → A, (a, s) as, such that one has a1 = a and a(st) = (as)t, for all a ∈ A and s, t ∈ S.

Ali Madanshekaf Semnan University

Preliminaries Weak Ideal Topology Results References Weak Topologies and Modal Closure Operators

Definition

Let S be a monoid with the identity 1. A (right) S-act is a set A equipped with an action µ : A × S → A, (a, s) as, such that one has a1 = a and a(st) = (as)t, for all a ∈ A and s, t ∈ S. We denote the category of right S-acts with equivariant maps between them by Act-S.

Ali Madanshekaf Semnan University

Preliminaries Weak Ideal Topology Results References Weak Topologies and Modal Closure Operators

Subobject classifier of Act-S

It is well known that the set Idl(S) of all right ideals K of S endowed with the action

Ali Madanshekaf Semnan University

Preliminaries Weak Ideal Topology Results References Weak Topologies and Modal Closure Operators

Subobject classifier of Act-S

It is well known that the set Idl(S) of all right ideals K of S endowed with the action K · s = {t ∈ S| st ∈ K},

Ali Madanshekaf Semnan University

Preliminaries Weak Ideal Topology Results References Weak Topologies and Modal Closure Operators

Subobject classifier of Act-S

It is well known that the set Idl(S) of all right ideals K of S endowed with the action K · s = {t ∈ S| st ∈ K}, and the monomorphism true : 1 = {θ} ֌ Idl(S), given by θ S, is the subobject classifier of Act-S.

Ali Madanshekaf Semnan University

Preliminaries Weak Ideal Topology Results References Weak Topologies and Modal Closure Operators

Subobject classifier of Act-S

It is well known that the set Idl(S) of all right ideals K of S endowed with the action K · s = {t ∈ S| st ∈ K}, and the monomorphism true : 1 = {θ} ֌ Idl(S), given by θ S, is the subobject classifier of Act-S. Indeed, associated to any subact B ⊆ A there is a unique map φB : A → Idl(S), so-called characteristic map of B ⊆ A, given by φB(a) = {s ∈ S| as ∈ B} such that the following diagram is a pullback B

• 1
• true
• A

φB Ω.

Ali Madanshekaf Semnan University

Preliminaries Weak Ideal Topology Results References Weak Topologies and Modal Closure Operators

Modal closure operator

A family C = (CA)A∈Act−S, where CA : SubAct−S(A) → SubAct−S(A), is called a modal closure operator on Act-S whenever for any S-act A we have

Ali Madanshekaf Semnan University

Preliminaries Weak Ideal Topology Results References Weak Topologies and Modal Closure Operators

Modal closure operator

A family C = (CA)A∈Act−S, where CA : SubAct−S(A) → SubAct−S(A), is called a modal closure operator on Act-S whenever for any S-act A we have

1

(Extensive) E ⊆ CA(E), for all subacts E of A;

Ali Madanshekaf Semnan University

Preliminaries Weak Ideal Topology Results References Weak Topologies and Modal Closure Operators

Modal closure operator

A family C = (CA)A∈Act−S, where CA : SubAct−S(A) → SubAct−S(A), is called a modal closure operator on Act-S whenever for any S-act A we have

1

(Extensive) E ⊆ CA(E), for all subacts E of A;

2

(Monotonicity) D ⊆ E implies CA(D) ⊆ CA(E), for all subacts D and E of A;

Ali Madanshekaf Semnan University

Preliminaries Weak Ideal Topology Results References Weak Topologies and Modal Closure Operators

Modal closure operator

A family C = (CA)A∈Act−S, where CA : SubAct−S(A) → SubAct−S(A), is called a modal closure operator on Act-S whenever for any S-act A we have

1

(Extensive) E ⊆ CA(E), for all subacts E of A;

2

(Monotonicity) D ⊆ E implies CA(D) ⊆ CA(E), for all subacts D and E of A;

3

(Modal) f −1(CB(E)) = CA(f −1(E)), for any equivariant map f : A → B and any subact E of B.

Ali Madanshekaf Semnan University

Preliminaries Weak Ideal Topology Results References Weak Topologies and Modal Closure Operators

Modal closure operator

A family C = (CA)A∈Act−S, where CA : SubAct−S(A) → SubAct−S(A), is called a modal closure operator on Act-S whenever for any S-act A we have

1

(Extensive) E ⊆ CA(E), for all subacts E of A;

2

(Monotonicity) D ⊆ E implies CA(D) ⊆ CA(E), for all subacts D and E of A;

3

(Modal) f −1(CB(E)) = CA(f −1(E)), for any equivariant map f : A → B and any subact E of B.

For simplicity the subact CA(B) of A is often denoted by B.

Ali Madanshekaf Semnan University

Preliminaries Weak Ideal Topology Results References Weak Topologies and Modal Closure Operators

Weak (Lawvere-Tierney) topology

An equivariant map j : Idl(S) → Idl(S) is said to be a weak (Lawvere-Tierney) topology on Act-S whenever the following hold:

Ali Madanshekaf Semnan University

Preliminaries Weak Ideal Topology Results References Weak Topologies and Modal Closure Operators

Weak (Lawvere-Tierney) topology

An equivariant map j : Idl(S) → Idl(S) is said to be a weak (Lawvere-Tierney) topology on Act-S whenever the following hold:

1

j(S) = S;

Ali Madanshekaf Semnan University

Preliminaries Weak Ideal Topology Results References Weak Topologies and Modal Closure Operators

Weak (Lawvere-Tierney) topology

An equivariant map j : Idl(S) → Idl(S) is said to be a weak (Lawvere-Tierney) topology on Act-S whenever the following hold:

1

j(S) = S;

2

j(K1 ∩ K2) ⊆ j(K1) ∩ j(K2), for all right ideals K1 and K2 of S.

Ali Madanshekaf Semnan University

Preliminaries Weak Ideal Topology Results References Weak Topologies and Modal Closure Operators

Weak (Lawvere-Tierney) topology

An equivariant map j : Idl(S) → Idl(S) is said to be a weak (Lawvere-Tierney) topology on Act-S whenever the following hold:

1

j(S) = S;

2

j(K1 ∩ K2) ⊆ j(K1) ∩ j(K2), for all right ideals K1 and K2 of S.

Furthermore, j is called productive whenever the inclusion in (2) is an equality.

Ali Madanshekaf Semnan University

Preliminaries Weak Ideal Topology Results References Weak Topologies and Modal Closure Operators

Weak (Lawvere-Tierney) topology

An equivariant map j : Idl(S) → Idl(S) is said to be a weak (Lawvere-Tierney) topology on Act-S whenever the following hold:

1

j(S) = S;

2

j(K1 ∩ K2) ⊆ j(K1) ∩ j(K2), for all right ideals K1 and K2 of S.

Furthermore, j is called productive whenever the inclusion in (2) is an equality. An idempotent weak topology on Act-S is called a topology

• n Act-S.

Ali Madanshekaf Semnan University

Preliminaries Weak Ideal Topology Results References Weak Topologies and Modal Closure Operators

Weak Topologies and Modal Closure Operators

Ali Madanshekaf Semnan University

Preliminaries Weak Ideal Topology Results References Weak Topologies and Modal Closure Operators

Weak Topologies and Modal Closure Operators

Corresponding to any weak topology j on Act-S, there is a modal closure operator on Act-S given by

Ali Madanshekaf Semnan University

Preliminaries Weak Ideal Topology Results References Weak Topologies and Modal Closure Operators

Weak Topologies and Modal Closure Operators

Corresponding to any weak topology j on Act-S, there is a modal closure operator on Act-S given by CA(B) = {a ∈ A| j(φB(a)) = S}, for any subact B ⊆ A.

Ali Madanshekaf Semnan University

Preliminaries Weak Ideal Topology Results References Weak Topologies and Modal Closure Operators

Weak Topologies and Modal Closure Operators

Corresponding to any weak topology j on Act-S, there is a modal closure operator on Act-S given by CA(B) = {a ∈ A| j(φB(a)) = S}, for any subact B ⊆ A. Conversely, any modal closure operator C on Act-S induces a weak topology j : Idl(S) → Idl(S) on Act-S given by

Ali Madanshekaf Semnan University

Preliminaries Weak Ideal Topology Results References Weak Topologies and Modal Closure Operators

Weak Topologies and Modal Closure Operators

Corresponding to any weak topology j on Act-S, there is a modal closure operator on Act-S given by CA(B) = {a ∈ A| j(φB(a)) = S}, for any subact B ⊆ A. Conversely, any modal closure operator C on Act-S induces a weak topology j : Idl(S) → Idl(S) on Act-S given by j(K) = {s ∈ S| K · s ∈ CIdl(S)({S})}.

Ali Madanshekaf Semnan University

Preliminaries Weak Ideal Topology Results References Weak Topologies and Modal Closure Operators

Weak Topologies and Modal Closure Operators

Corresponding to any weak topology j on Act-S, there is a modal closure operator on Act-S given by CA(B) = {a ∈ A| j(φB(a)) = S}, for any subact B ⊆ A. Conversely, any modal closure operator C on Act-S induces a weak topology j : Idl(S) → Idl(S) on Act-S given by j(K) = {s ∈ S| K · s ∈ CIdl(S)({S})}. One can easily check that Lemma On the topos Act-S, weak topologies j are in one-to-one correspondence with modal closure operators (·).

Ali Madanshekaf Semnan University

Preliminaries Weak Ideal Topology Results References Weak Topologies and Modal Closure Operators

j-closed, j-dense, j-sheaf and j-separated

Let j be a weak topology on Act-S and (·) the modal closure

• perator associated to j. A subact B of any act A is said to be

Ali Madanshekaf Semnan University

Preliminaries Weak Ideal Topology Results References Weak Topologies and Modal Closure Operators

j-closed, j-dense, j-sheaf and j-separated

Let j be a weak topology on Act-S and (·) the modal closure

• perator associated to j. A subact B of any act A is said to be

j-closed (in A) if B = B and it is j-dense (in A) if B = A. An S-act C is called: a j-sheaf whenever for any j-dense subact B ⊆ A, one can uniquely extend any equivariant map h : B → C to A, i.e., g|B = h

Ali Madanshekaf Semnan University

Preliminaries Weak Ideal Topology Results References Weak Topologies and Modal Closure Operators

j-closed, j-dense, j-sheaf and j-separated

Let j be a weak topology on Act-S and (·) the modal closure

• perator associated to j. A subact B of any act A is said to be

j-closed (in A) if B = B and it is j-dense (in A) if B = A. An S-act C is called: a j-sheaf whenever for any j-dense subact B ⊆ A, one can uniquely extend any equivariant map h : B → C to A, i.e., g|B = h B

• h
• A

g

• C

Ali Madanshekaf Semnan University

Preliminaries Weak Ideal Topology Results References Weak Topologies and Modal Closure Operators

j-closed, j-dense, j-sheaf and j-separated

Let j be a weak topology on Act-S and (·) the modal closure

• perator associated to j. A subact B of any act A is said to be

j-closed (in A) if B = B and it is j-dense (in A) if B = A. An S-act C is called: a j-sheaf whenever for any j-dense subact B ⊆ A, one can uniquely extend any equivariant map h : B → C to A, i.e., g|B = h B

• h
• A

g

• C

a j-separated act if the arrow g exists, it is unique.

Ali Madanshekaf Semnan University

Preliminaries Weak Ideal Topology Results References Different names of the weak ideal topology in literature Separated and I-separated Categories of jI -separated acts An act of equivariant maps

Ideal closure operator and weak ideal topology

Let I be a left ideal of S.

Ali Madanshekaf Semnan University

Preliminaries Weak Ideal Topology Results References Different names of the weak ideal topology in literature Separated and I-separated Categories of jI -separated acts An act of equivariant maps

Ideal closure operator and weak ideal topology

Let I be a left ideal of S. Then the ideal (or residual) closure

• perator with respect to I, C I, is defined by

Ali Madanshekaf Semnan University

Preliminaries Weak Ideal Topology Results References Different names of the weak ideal topology in literature Separated and I-separated Categories of jI -separated acts An act of equivariant maps

Ideal closure operator and weak ideal topology

Let I be a left ideal of S. Then the ideal (or residual) closure

• perator with respect to I, C I, is defined by

C I

A(B) = {a ∈ A| ∀t ∈ I, at ∈ B},

for any subact B ⊆ A.

Ali Madanshekaf Semnan University

Preliminaries Weak Ideal Topology Results References Different names of the weak ideal topology in literature Separated and I-separated Categories of jI -separated acts An act of equivariant maps

Ideal closure operator and weak ideal topology

Let I be a left ideal of S. Then the ideal (or residual) closure

• perator with respect to I, C I, is defined by

C I

A(B) = {a ∈ A| ∀t ∈ I, at ∈ B},

for any subact B ⊆ A. By a weak ideal topology on Act-S (induced by I) we mean the equivariant map jI : Idl(S) → Idl(S) given by

Ali Madanshekaf Semnan University

Preliminaries Weak Ideal Topology Results References Different names of the weak ideal topology in literature Separated and I-separated Categories of jI -separated acts An act of equivariant maps

Ideal closure operator and weak ideal topology

Let I be a left ideal of S. Then the ideal (or residual) closure

• perator with respect to I, C I, is defined by

C I

A(B) = {a ∈ A| ∀t ∈ I, at ∈ B},

for any subact B ⊆ A. By a weak ideal topology on Act-S (induced by I) we mean the equivariant map jI : Idl(S) → Idl(S) given by jI(K) = {s ∈ S| ∀t ∈ I, st ∈ K}, for any K ∈ Idl(S).

Ali Madanshekaf Semnan University

Preliminaries Weak Ideal Topology Results References Different names of the weak ideal topology in literature Separated and I-separated Categories of jI -separated acts An act of equivariant maps

Ideal closure operator and weak ideal topology

Let I be a left ideal of S. Then the ideal (or residual) closure

• perator with respect to I, C I, is defined by

C I

A(B) = {a ∈ A| ∀t ∈ I, at ∈ B},

for any subact B ⊆ A. By a weak ideal topology on Act-S (induced by I) we mean the equivariant map jI : Idl(S) → Idl(S) given by jI(K) = {s ∈ S| ∀t ∈ I, st ∈ K}, for any K ∈ Idl(S). Lemma For a (left) ideal I of S, jI is a topology on Act-S iff I 2 = I ((IS)2 = IS).

Ali Madanshekaf Semnan University

Preliminaries Weak Ideal Topology Results References Different names of the weak ideal topology in literature Separated and I-separated Categories of jI -separated acts An act of equivariant maps

Other terminologies

Weak ideal topology and related notions have been called by different names in other contexts. For example

1 The theory of automata: M-filter Ali Madanshekaf Semnan University

Preliminaries Weak Ideal Topology Results References Different names of the weak ideal topology in literature Separated and I-separated Categories of jI -separated acts An act of equivariant maps

Other terminologies

Weak ideal topology and related notions have been called by different names in other contexts. For example

1 The theory of automata: M-filter

´ Ciri´ c, M. (1999). Lattices of subautomata and direct sum decompositions of

• automata. Algebra Colloq. 6(1):71-88.

Ali Madanshekaf Semnan University

Preliminaries Weak Ideal Topology Results References Different names of the weak ideal topology in literature Separated and I-separated Categories of jI -separated acts An act of equivariant maps

Other terminologies

Weak ideal topology and related notions have been called by different names in other contexts. For example

1 The theory of automata: M-filter

´ Ciri´ c, M. (1999). Lattices of subautomata and direct sum decompositions of

• automata. Algebra Colloq. 6(1):71-88.

2 Semigroup theory: Sequentially closure operator or s-closure

• perator

Ali Madanshekaf Semnan University

Preliminaries Weak Ideal Topology Results References Different names of the weak ideal topology in literature Separated and I-separated Categories of jI -separated acts An act of equivariant maps

Other terminologies

Weak ideal topology and related notions have been called by different names in other contexts. For example

1 The theory of automata: M-filter

´ Ciri´ c, M. (1999). Lattices of subautomata and direct sum decompositions of

• automata. Algebra Colloq. 6(1):71-88.

2 Semigroup theory: Sequentially closure operator or s-closure

• perator

Mahmoudi, M., Moghaddasi Angizan, Gh. (2007). Sequentially injective hull of acts over idempotent semigroups. Semigroup Forum 74:240-246.

Ali Madanshekaf Semnan University

Preliminaries Weak Ideal Topology Results References Different names of the weak ideal topology in literature Separated and I-separated Categories of jI -separated acts An act of equivariant maps

Other terminologies

Weak ideal topology and related notions have been called by different names in other contexts. For example

1 The theory of automata: M-filter

´ Ciri´ c, M. (1999). Lattices of subautomata and direct sum decompositions of

• automata. Algebra Colloq. 6(1):71-88.

2 Semigroup theory: Sequentially closure operator or s-closure

• perator

Mahmoudi, M., Moghaddasi Angizan, Gh. (2007). Sequentially injective hull of acts over idempotent semigroups. Semigroup Forum 74:240-246.

3 Projection algebras: m-closure operator and m-separated

algebra

Ali Madanshekaf Semnan University

Preliminaries Weak Ideal Topology Results References Different names of the weak ideal topology in literature Separated and I-separated Categories of jI -separated acts An act of equivariant maps

Other terminologies

Weak ideal topology and related notions have been called by different names in other contexts. For example

1 The theory of automata: M-filter

´ Ciri´ c, M. (1999). Lattices of subautomata and direct sum decompositions of

• automata. Algebra Colloq. 6(1):71-88.

2 Semigroup theory: Sequentially closure operator or s-closure

• perator

Mahmoudi, M., Moghaddasi Angizan, Gh. (2007). Sequentially injective hull of acts over idempotent semigroups. Semigroup Forum 74:240-246.

3 Projection algebras: m-closure operator and m-separated

algebra

Ehrig, H., Herrlich, H. (1988). The construct PRO of projection spaces: Its internal structure. Lect. Notes Comput. Sci. 393:286-293. Giuli, E. (1994). On m-separated projection spaces. Appl. Categ. Struct. 2:91-99.

Ali Madanshekaf Semnan University

Preliminaries Weak Ideal Topology Results References Different names of the weak ideal topology in literature Separated and I-separated Categories of jI -separated acts An act of equivariant maps

Illustration

Here, we will explain some of them:

Ali Madanshekaf Semnan University

Preliminaries Weak Ideal Topology Results References Different names of the weak ideal topology in literature Separated and I-separated Categories of jI -separated acts An act of equivariant maps

Illustration

Here, we will explain some of them: Let M be a monoid which has no (non-trivial) left invertible element, i.e., S = M\{1} is a left ideal of M. Then:

Ali Madanshekaf Semnan University

Preliminaries Weak Ideal Topology Results References Different names of the weak ideal topology in literature Separated and I-separated Categories of jI -separated acts An act of equivariant maps

Illustration

Here, we will explain some of them: Let M be a monoid which has no (non-trivial) left invertible element, i.e., S = M\{1} is a left ideal of M. Then:

Ali Madanshekaf Semnan University

Preliminaries Weak Ideal Topology Results References Different names of the weak ideal topology in literature Separated and I-separated Categories of jI -separated acts An act of equivariant maps

Illustration

Here, we will explain some of them: Let M be a monoid which has no (non-trivial) left invertible element, i.e., S = M\{1} is a left ideal of M. Then: any jS-closed subact of any M-act A is called an M-filter subact of A.

Ali Madanshekaf Semnan University

Preliminaries Weak Ideal Topology Results References Different names of the weak ideal topology in literature Separated and I-separated Categories of jI -separated acts An act of equivariant maps

Illustration

Here, we will explain some of them: Let M be a monoid which has no (non-trivial) left invertible element, i.e., S = M\{1} is a left ideal of M. Then: any jS-closed subact of any M-act A is called an M-filter subact of A. The s-closure operator on Act-M is a modal closure operator in our sense which gives us exactly the weak ideal topology jS

• n Act-M induced by the ideal S of M.

Ali Madanshekaf Semnan University

Preliminaries Weak Ideal Topology Results References Different names of the weak ideal topology in literature Separated and I-separated Categories of jI -separated acts An act of equivariant maps

Projection algebras

Act-N∞ Let (N∞ = N ∪ {∞}, min) be the monoid of extended natural

• numbers. Then the topos Act-N∞ often is called the category of

projection algebras and is denoted by PRO. For any natural number m, the m-closure operator defined on objects of PRO by Giuli, E. (1994) is a modal closure operator in our sense which gives precisely the ideal topology jI on Act-N∞ induced by the ideal I =↓ m(= {n ∈ N | n ≤ m}) of the idempotent monoid N∞. Note that if I = N, then jI stands for the topology associated with the s-closure operator on Act-N∞.

Ali Madanshekaf Semnan University

Preliminaries Weak Ideal Topology Results References Different names of the weak ideal topology in literature Separated and I-separated Categories of jI -separated acts An act of equivariant maps

Separated act

Recall that for a semigroup S, an S-act A is called separated if for all a, b ∈ A,

Ali Madanshekaf Semnan University

Preliminaries Weak Ideal Topology Results References Different names of the weak ideal topology in literature Separated and I-separated Categories of jI -separated acts An act of equivariant maps

Separated act

Recall that for a semigroup S, an S-act A is called separated if for all a, b ∈ A, as = bs, ∀s ∈ S = ⇒ a = b. (1)

Ali Madanshekaf Semnan University

Preliminaries Weak Ideal Topology Results References Different names of the weak ideal topology in literature Separated and I-separated Categories of jI -separated acts An act of equivariant maps

Separated act

Recall that for a semigroup S, an S-act A is called separated if for all a, b ∈ A, as = bs, ∀s ∈ S = ⇒ a = b. (1) Here we may consider the semigroup S as an ideal (of course, two-sided) of the induced monoid S1. We can extend this definition to an arbitrary left ideal I of an arbitrary monoid as follows:

Ali Madanshekaf Semnan University

Preliminaries Weak Ideal Topology Results References Different names of the weak ideal topology in literature Separated and I-separated Categories of jI -separated acts An act of equivariant maps

Separated act

Recall that for a semigroup S, an S-act A is called separated if for all a, b ∈ A, as = bs, ∀s ∈ S = ⇒ a = b. (1) Here we may consider the semigroup S as an ideal (of course, two-sided) of the induced monoid S1. We can extend this definition to an arbitrary left ideal I of an arbitrary monoid as follows: I-separated act Let I be a left ideal of a monoid S. An S-act A is said to be I-separated provided

Ali Madanshekaf Semnan University

Preliminaries Weak Ideal Topology Results References Different names of the weak ideal topology in literature Separated and I-separated Categories of jI -separated acts An act of equivariant maps

Separated act

Recall that for a semigroup S, an S-act A is called separated if for all a, b ∈ A, as = bs, ∀s ∈ S = ⇒ a = b. (1) Here we may consider the semigroup S as an ideal (of course, two-sided) of the induced monoid S1. We can extend this definition to an arbitrary left ideal I of an arbitrary monoid as follows: I-separated act Let I be a left ideal of a monoid S. An S-act A is said to be I-separated provided as = bs, ∀s ∈ I = ⇒ a = b, for all a, b ∈ A.

Ali Madanshekaf Semnan University

Preliminaries Weak Ideal Topology Results References Different names of the weak ideal topology in literature Separated and I-separated Categories of jI -separated acts An act of equivariant maps

Some famous categories of jI-separated acts

1 The category PROs of all separated projection algebras is

exactly the category SepjI (Act − N∞) for the idempotent ideal I = N of the monoid N∞.

Ali Madanshekaf Semnan University

Preliminaries Weak Ideal Topology Results References Different names of the weak ideal topology in literature Separated and I-separated Categories of jI -separated acts An act of equivariant maps

Some famous categories of jI-separated acts

1 The category PROs of all separated projection algebras is

exactly the category SepjI (Act − N∞) for the idempotent ideal I = N of the monoid N∞.

2 Let M be a monoid which has no (non-trivial) left invertible

• element. Then the category M-Setss of all separated M-sets

defined by some authors is exactly the category SepjS(Act − M).

Ali Madanshekaf Semnan University

Preliminaries Weak Ideal Topology Results References Different names of the weak ideal topology in literature Separated and I-separated Categories of jI -separated acts An act of equivariant maps

I-separated and jI-separated

Lemma For a left ideal I of S, an S-act A is I-separated iff it is jI-separated.

Ali Madanshekaf Semnan University

Preliminaries Weak Ideal Topology Results References Different names of the weak ideal topology in literature Separated and I-separated Categories of jI -separated acts An act of equivariant maps

I-separated and jI-separated

Lemma For a left ideal I of S, an S-act A is I-separated iff it is jI-separated. From now on, we use the notion I-separated instead of jI-separated.

Ali Madanshekaf Semnan University

Preliminaries Weak Ideal Topology Results References Different names of the weak ideal topology in literature Separated and I-separated Categories of jI -separated acts An act of equivariant maps

EI(A)

Ali Madanshekaf Semnan University

Preliminaries Weak Ideal Topology Results References Different names of the weak ideal topology in literature Separated and I-separated Categories of jI -separated acts An act of equivariant maps

EI(A)

Let I be a central band of S. We denote EI(A) for the S-act consisting of all equivariant maps f : I → A endowed with the action

Ali Madanshekaf Semnan University

Preliminaries Weak Ideal Topology Results References Different names of the weak ideal topology in literature Separated and I-separated Categories of jI -separated acts An act of equivariant maps

EI(A)

Let I be a central band of S. We denote EI(A) for the S-act consisting of all equivariant maps f : I → A endowed with the action (f · m)(n) = f (n)m, ∀m ∈ S and n ∈ I.

Ali Madanshekaf Semnan University

Preliminaries Weak Ideal Topology Results References Different names of the weak ideal topology in literature Separated and I-separated Categories of jI -separated acts An act of equivariant maps

EI(A)

Let I be a central band of S. We denote EI(A) for the S-act consisting of all equivariant maps f : I → A endowed with the action (f · m)(n) = f (n)m, ∀m ∈ S and n ∈ I. The following provides major property of EI(A):

Ali Madanshekaf Semnan University

Preliminaries Weak Ideal Topology Results References Different names of the weak ideal topology in literature Separated and I-separated Categories of jI -separated acts An act of equivariant maps

EI(A)

Let I be a central band of S. We denote EI(A) for the S-act consisting of all equivariant maps f : I → A endowed with the action (f · m)(n) = f (n)m, ∀m ∈ S and n ∈ I. The following provides major property of EI(A): Lemma For any S-act A, the S-act EI(A) is I-separated. Also, it is jI-injective and so jI-sheaf if A is I-separated.

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Preliminaries Weak Ideal Topology Results References

One part of the associated sheaf functor w.r.t. jI

Ali Madanshekaf Semnan University

Preliminaries Weak Ideal Topology Results References

One part of the associated sheaf functor w.r.t. jI

Let I be a central band of S. Then the inclusion functor SepjI (Act − S) ֒ → Act − S has a left adjoint

Ali Madanshekaf Semnan University

Preliminaries Weak Ideal Topology Results References

One part of the associated sheaf functor w.r.t. jI

Let I be a central band of S. Then the inclusion functor SepjI (Act − S) ֒ → Act − S has a left adjoint L : Act − S − → SepjI (Act − S)

Ali Madanshekaf Semnan University

Preliminaries Weak Ideal Topology Results References

One part of the associated sheaf functor w.r.t. jI

Let I be a central band of S. Then the inclusion functor SepjI (Act − S) ֒ → Act − S has a left adjoint L : Act − S − → SepjI (Act − S) defined by L(A) = A/σA in which σA stands for the following congruence on A σA = {(a, b) ∈ A × A| ∀t ∈ I, at = bt}.

Ali Madanshekaf Semnan University

Preliminaries Weak Ideal Topology Results References

Another part of the associated sheaf functor w.r.t. jI

The next result provides the associated sheaf functor with respect to the ideal topology jI on Act-S.

Ali Madanshekaf Semnan University

Preliminaries Weak Ideal Topology Results References

Another part of the associated sheaf functor w.r.t. jI

The next result provides the associated sheaf functor with respect to the ideal topology jI on Act-S. Theorem Let I be a central band of S. The inclusion functor ι : ShjI (Act − S) ֒ → SepjI (Act − S) has a left adjoint

Ali Madanshekaf Semnan University

Preliminaries Weak Ideal Topology Results References

Another part of the associated sheaf functor w.r.t. jI

The next result provides the associated sheaf functor with respect to the ideal topology jI on Act-S. Theorem Let I be a central band of S. The inclusion functor ι : ShjI (Act − S) ֒ → SepjI (Act − S) has a left adjoint EI : SepjI (Act − S) − → ShjI (Act − S)

Ali Madanshekaf Semnan University

Preliminaries Weak Ideal Topology Results References

Another part of the associated sheaf functor w.r.t. jI

The next result provides the associated sheaf functor with respect to the ideal topology jI on Act-S. Theorem Let I be a central band of S. The inclusion functor ι : ShjI (Act − S) ֒ → SepjI (Act − S) has a left adjoint EI : SepjI (Act − S) − → ShjI (Act − S) which assigns to any I-separated S-act A the jI-sheaf EI(A) and to any equivariant map f : A → B the equivariant map f∗ : EI(A) → EI(B) given by composition with f .

Ali Madanshekaf Semnan University

Preliminaries Weak Ideal Topology Results References

The associated sheaf functor

Combining the last two results gives

Ali Madanshekaf Semnan University

Preliminaries Weak Ideal Topology Results References

The associated sheaf functor

Combining the last two results gives Let I be a central band of S. Then, the inclusion functor ShjI (Act − S) ֒ → Act − S has a left adjoint, called the associated sheaf functor,

Ali Madanshekaf Semnan University

Preliminaries Weak Ideal Topology Results References

The associated sheaf functor

Combining the last two results gives Let I be a central band of S. Then, the inclusion functor ShjI (Act − S) ֒ → Act − S has a left adjoint, called the associated sheaf functor, a : Act − S − → ShjI (Act − S)

Ali Madanshekaf Semnan University

Preliminaries Weak Ideal Topology Results References

The associated sheaf functor

Combining the last two results gives Let I be a central band of S. Then, the inclusion functor ShjI (Act − S) ֒ → Act − S has a left adjoint, called the associated sheaf functor, a : Act − S − → ShjI (Act − S) defined by a(A) = EI(A/σA) in which σA stands for the following congruence on A σA = {(a, b) ∈ A × A| ∀t ∈ I, at = bt}.

Ali Madanshekaf Semnan University

Preliminaries Weak Ideal Topology Results References

Ideal topology and the implication topology

Ali Madanshekaf Semnan University

Preliminaries Weak Ideal Topology Results References

Ideal topology and the implication topology

Recall that Idl(S) has a frame structure.

Ali Madanshekaf Semnan University

Preliminaries Weak Ideal Topology Results References

Ideal topology and the implication topology

Recall that Idl(S) has a frame structure. For any two right ideals I and K of S, one has

Ali Madanshekaf Semnan University

Preliminaries Weak Ideal Topology Results References

Ideal topology and the implication topology

Recall that Idl(S) has a frame structure. For any two right ideals I and K of S, one has I ⇒ K = {s ∈ S| for all t ∈ S; if st ∈ I then st ∈ K}.

Ali Madanshekaf Semnan University

Preliminaries Weak Ideal Topology Results References

Ideal topology and the implication topology

Recall that Idl(S) has a frame structure. For any two right ideals I and K of S, one has I ⇒ K = {s ∈ S| for all t ∈ S; if st ∈ I then st ∈ K}. The following gives another structure of the ideal topology jI on Act-S.

Ali Madanshekaf Semnan University

Preliminaries Weak Ideal Topology Results References

Ideal topology and the implication topology

Recall that Idl(S) has a frame structure. For any two right ideals I and K of S, one has I ⇒ K = {s ∈ S| for all t ∈ S; if st ∈ I then st ∈ K}. The following gives another structure of the ideal topology jI on Act-S. Proposition Let I be a central band of S. Then, the two topologies I ⇒ (−) and jI on Act-S coincide.

Ali Madanshekaf Semnan University

Preliminaries Weak Ideal Topology Results References

References

• D. Dikranjan and W. Tholen,

Categorical Structure of Closure Operators, Kluwer, Netherlands, (1995).

• S. N. Hosseini and S. SH. Mousavi,

A Relation Between Closure Operators on a Small Category and Its Category of Presheaves,

• Appl. Categ. Struc., 14 (2006), 99-110.
• M. Kilp, U. Knauer and A.V. Mikhalev,

Monoids, Acts and Categories, Walter de Gruyter, Berlin, (2000).

• S. Mac Lane and I. Moerdijk,

Sheaves in Geometry and Logic, Springer-Verlag, New York, (1992).

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Preliminaries Weak Ideal Topology Results References

Thank You for Your Attention!

Ali Madanshekaf Semnan University