Quasi Injectivity of Partially Ordered Acts

Quasi injectivity of partially ordered acts

Mahdieh Yavari Shahid Quasi injectivity of partially ordered acts Beheshti University

Introduction Mahdieh Yavari Preliminaries Shahid Beheshti University E-quasi injectivity in Pos-S

E-quasi injectivity and TACL 26-30 June 2017 θ-extensions Charles University Reversiblity and E-quasi injectivity of S-posets

References

1/50 ⇒

Partially Ordered Sets with actions of a pomonoid S on them (Pos-S)

Introduction

Quasi injectivity of partially ordered acts Mahdieh Actions of a Monoid on Sets Yavari Shahid Beheshti University +

Introduction Actions of a Monoid on Partially Ordered Sets Preliminaries

E-quasi injectivity in Pos-S

E-quasi injectivity and θ-extensions

Reversiblity and E-quasi injectivity of S-posets

References

2/50 Introduction

Quasi injectivity of partially ordered acts Mahdieh Actions of a Monoid on Sets Yavari Shahid Beheshti University +

Introduction Actions of a Monoid on Partially Ordered Sets Preliminaries

E-quasi injectivity in ⇒ Pos-S

E-quasi injectivity and Partially Ordered Sets with actions of a pomonoid S θ-extensions on them (Pos-S) Reversiblity and E-quasi injectivity of S-posets

References

2/50 Banaschewski: In the category of posets with order preserving maps (Pos)

E-injective = Complete posets

Introduction

Quasi injectivity of partially ordered acts

Mahdieh Injectivity Yavari Shahid Injectivity, which is about extending morphisms to a bigger Beheshti University domain of deﬁnition, plays a fundamental role in many branches of mathematics. Introduction

Preliminaries

E-quasi injectivity in Pos-S

E-quasi injectivity and θ-extensions

Reversiblity and E-quasi injectivity of S-posets

References

3/50 Introduction

Quasi injectivity of partially ordered acts

Preliminaries

E-quasi injectivity in Pos-S Banaschewski: E-quasi In the category of posets with order preserving maps (Pos) injectivity and θ-extensions E-injective = Complete posets Reversiblity and E-quasi injectivity of S-posets

References

3/50 Ebrahimi, Mahmoudi, Rasouli: In Pos-S Mono-injective object = Trivial object Pos-S has enough E-injective objects.

Introduction

Quasi injectivity of partially ordered acts Mahdieh Sikorski: Yavari Shahid Beheshti In the category of BoolAlg University E-injective = Complete Boolean algebras Introduction

Preliminaries

E-quasi injectivity in Pos-S

E-quasi injectivity and θ-extensions

Reversiblity and E-quasi injectivity of S-posets

References

4/50 Introduction

Quasi injectivity of partially ordered acts Mahdieh Sikorski: Yavari Shahid Beheshti In the category of BoolAlg University E-injective = Complete Boolean algebras Introduction

Preliminaries

E-quasi injectivity in Pos-S Ebrahimi, Mahmoudi, Rasouli: E-quasi In Pos-S Mono-injective object = Trivial object injectivity and θ-extensions Pos-S has enough E-injective objects. Reversiblity and E-quasi injectivity of S-posets

References

4/50 Quasi injective S-acts have been studied by Berthiaume, Satyanarayana, Lopez and Luedeman. Ahsan, gave a characterization of monoids whose class of quasi injective S-acts is closed under the formation of direct sums. Also, he investigated monoids all of whose S-acts are quasi injective. Roueentan and Ershad, introduced duo S-act, inspired from the concept of duo module in module theory, which is tightly related to quasi injectivity.

Quasi Injectivity

Quasi injectivity of partially ordered acts The study of diﬀerent kinds of weakly injectivity is an

Mahdieh interesting subject for mathematicians. One of the Yavari Shahid important kinds of weakly injectivity is quasi injectivity. Beheshti University

Introduction

Preliminaries

E-quasi injectivity in Pos-S

E-quasi injectivity and θ-extensions

Reversiblity and E-quasi injectivity of S-posets

References

5/50 Ahsan, gave a characterization of monoids whose class of quasi injective S-acts is closed under the formation of direct sums. Also, he investigated monoids all of whose S-acts are quasi injective. Roueentan and Ershad, introduced duo S-act, inspired from the concept of duo module in module theory, which is tightly related to quasi injectivity.

Quasi Injectivity

Quasi injectivity of partially ordered acts The study of diﬀerent kinds of weakly injectivity is an

Mahdieh interesting subject for mathematicians. One of the Yavari Shahid important kinds of weakly injectivity is quasi injectivity. Beheshti University Quasi injective S-acts have been studied by Berthiaume, Satyanarayana, Lopez and Luedeman. Introduction

Preliminaries

E-quasi injectivity in Pos-S

E-quasi injectivity and θ-extensions

Reversiblity and E-quasi injectivity of S-posets

References

5/50 Roueentan and Ershad, introduced duo S-act, inspired from the concept of duo module in module theory, which is tightly related to quasi injectivity.

Quasi Injectivity

Quasi injectivity of partially ordered acts The study of diﬀerent kinds of weakly injectivity is an

Reversiblity and E-quasi injectivity of S-posets

References

5/50 Quasi Injectivity

Quasi injectivity of partially ordered acts The study of diﬀerent kinds of weakly injectivity is an

Preliminaries Ahsan, gave a characterization of monoids whose class of E-quasi quasi injective S-acts is closed under the formation of injectivity in Pos-S direct sums. Also, he investigated monoids all of whose E-quasi S-acts are quasi injective. injectivity and θ-extensions Roueentan and Ershad, introduced duo S-act, inspired Reversiblity and E-quasi from the concept of duo module in module theory, which injectivity of S-posets is tightly related to quasi injectivity.

References

5/50 Preliminaries

Quasi injectivity of partially ordered acts Mahdieh Deﬁnition (Act-S) Yavari Shahid Beheshti Recall that a right S-act or S-set for a monoid S is a set A University equipped with an action A × S → A,(a, s) as, such that Introduction (1) ae = a (e is the identity element of S), Preliminaries (2) a(st) = (as)t, for all a ∈ A and s, t ∈ S. E-quasi injectivity in Pos-S Let Act-S denote the category of all S-acts with action E-quasi preserving maps (f : A → B with f (as) = f (a)s, for all injectivity and θ-extensions a ∈ A, s ∈ S). Reversiblity and E-quasi injectivity of S-posets

References

6/50 Preliminaries

Quasi injectivity of partially ordered acts

Mahdieh Yavari Shahid Beheshti University Deﬁnition (Zero element) Introduction Let A be an S-act. An element a ∈ A is called a zero, ﬁxed, or Preliminaries a trap element if as = a, for all s ∈ S. E-quasi injectivity in Pos-S

E-quasi injectivity and θ-extensions

Reversiblity and E-quasi injectivity of S-posets

References

7/50 Deﬁnition (Right (Left) Zero Semigroup) A semigroup S is called right (left) zero if its multiplication is deﬁned by st = t(st = s), for all s, t ∈ S. Also, S∪{˙ e} is called a right (left) zero semigroup with an adjoined identity, if S is a right (left) zero semigroup and se = s = es, for all s ∈ S, and ee = e.

Preliminaries

Quasi injectivity of partially ordered acts Deﬁnition (Po-monoid, Po-group) Mahdieh A po-monoid (po-group) S is a monoid (group) with a partial Yavari Shahid order ≤ which is compatible with its binary operation (that is, Beheshti 0 0 0 0 0 0 University for s, t, s , t ∈ S, s ≤ t and s ≤ t imply ss ≤ tt ).

Introduction

Preliminaries

E-quasi injectivity in Pos-S

E-quasi injectivity and θ-extensions

Reversiblity and E-quasi injectivity of S-posets

References

8/50 Preliminaries

Introduction

Preliminaries

E-quasi Deﬁnition (Right (Left) Zero Semigroup) injectivity in Pos-S A semigroup S is called right (left) zero if its multiplication is E-quasi ˙ injectivity and deﬁned by st = t(st = s), for all s, t ∈ S. Also, S∪{e} is θ-extensions called a right (left) zero semigroup with an adjoined identity, if Reversiblity and E-quasi S is a right (left) zero semigroup and se = s = es, for all injectivity of S-posets s ∈ S, and ee = e.

References

8/50 Preliminaries

Quasi injectivity of partially ordered acts

Mahdieh Yavari Shahid Beheshti Deﬁnition (Pos-S) University For a po-monoid S, a right S-poset is a poset A which is also Introduction an S-act whose action λ : A × S → A is order-preserving, Preliminaries where A × S is considered as a poset with componentwise E-quasi injectivity in order. The category of all S-posets with action preserving Pos-S monotone maps between them is denoted by Pos-S. E-quasi injectivity and θ-extensions

Reversiblity and E-quasi injectivity of S-posets

References

9/50 Preliminaries

Quasi injectivity of partially ordered acts

Mahdieh Yavari Shahid Beheshti Deﬁnition (Order-embedding) University A morphism f : A → B in the category Pos-S and its Introduction subcategories, is called order-embedding, or brieﬂy embedding, Preliminaries if for all x, y ∈ A, f (x) ≤ f (y) if and only if x ≤ y. E-quasi injectivity in We consider E to be the class of all embeddings in the category Pos-S Pos-S and its subcategories. E-quasi injectivity and θ-extensions

Reversiblity and E-quasi injectivity of S-posets

References

10/50 Relation between Monomorphisms and Embeddings

Quasi injectivity of partially ordered acts Mahdieh In the category Pos-S : Yavari Shahid Embedding ⇒ Mono Beheshti University Mono 6⇒ Embedding

Introduction

Preliminaries Suppose that S is an arbitrary po-monoid. Deﬁne f by 0 E-quasi f (⊥) = 0, f (a) = 1 and f (a ) = 2. injectivity in Pos-S a a0 2 E-quasi @ f 1 injectivity and r@ r → r θ-extensions @ r Reversiblity and E-quasi ⊥r 0r injectivity of S-posets

References A B

11/50 Preliminaries

Quasi injectivity of partially ordered acts Deﬁnition (M-injective) Mahdieh Yavari For a class M of monomorphisms in a category C, an object Shahid Beheshti A ∈ C is called M-injective if for each M-morphism University f : B → C and morphism g : B → A there exists a morphism Introduction h : C → A such that hf = g. Preliminaries E-quasi f ∈M injectivity in B / C Pos-S

E-quasi g h injectivity and θ-extensions A Reversiblity and E-quasi injectivity of S-posets

References

12/50 Preliminaries

Quasi injectivity of partially ordered acts

Mahdieh Deﬁnition (M-quasi injective) Yavari Shahid An object A ∈ C is called M-quasi injective if for each Beheshti University M-morphism m : B → A and any morphism f : B → A there exists a morphism f¯ : A → A which extends f . Introduction

Preliminaries m∈M E-quasi B / A injectivity in Pos-S f f¯ E-quasi injectivity and A θ-extensions

Reversiblity and E-quasi injectivity of S-posets

References

13/50 Preliminaries

Quasi injectivity of partially ordered acts Deﬁnition (M-absolute retract) Mahdieh Yavari An object A ∈ C is called M-absolute retract if it is a retract Shahid Beheshti of each of its M-extensions; that is, for each M-morphism University f : A → C there exists a morphism h : C → A such that Introduction hf = idA, in which case h is said to be a retraction. Preliminaries E-quasi f ∈M injectivity in A / C Pos-S

E-quasi idA h injectivity and θ-extensions A Reversiblity and E-quasi injectivity of S-posets

References

14/50 But the converse of above theorem, is not true in general. For the converse, we need some conditions.

Theorem In the category Pos-S, E-injective ⇐⇒E -absolute retract

Preliminaries

Quasi injectivity of partially Theorem ordered acts Mahdieh M-injective =⇒M -absolute retract Yavari Shahid Beheshti University

Introduction

Preliminaries

E-quasi injectivity in Pos-S

E-quasi injectivity and θ-extensions

Reversiblity and E-quasi injectivity of S-posets

References

15/50 Theorem In the category Pos-S, E-injective ⇐⇒E -absolute retract

Preliminaries

Quasi injectivity of partially Theorem ordered acts Mahdieh M-injective =⇒M -absolute retract Yavari Shahid Beheshti University

Introduction But the converse of above theorem, is not true in general. For Preliminaries the converse, we need some conditions. E-quasi injectivity in Pos-S

E-quasi injectivity and θ-extensions

Reversiblity and E-quasi injectivity of S-posets

References

15/50 Preliminaries

Quasi injectivity of partially Theorem ordered acts Mahdieh M-injective =⇒M -absolute retract Yavari Shahid Beheshti University

Introduction But the converse of above theorem, is not true in general. For Preliminaries the converse, we need some conditions. E-quasi injectivity in Pos-S

E-quasi injectivity and θ-extensions Theorem Reversiblity and E-quasi In the category Pos-S, injectivity of S-posets E-injective ⇐⇒E -absolute retract

References

15/50 E-injective 6⇐E -quasi injective

Theorem In the category Pos-S, Complete posets with identity actions ⇒E -quasi injective

Some results about an E-quasi injective S-poset

Quasi injectivity of partially ordered acts Theorem Mahdieh Yavari Shahid In the category Pos-S, Beheshti University E-injective ⇒E -quasi injective

Introduction

Preliminaries

E-quasi injectivity in Pos-S

E-quasi injectivity and θ-extensions

Reversiblity and E-quasi injectivity of S-posets

References

16/50 Theorem In the category Pos-S, Complete posets with identity actions ⇒E -quasi injective

Some results about an E-quasi injective S-poset

Quasi injectivity of partially ordered acts Theorem Mahdieh Yavari Shahid In the category Pos-S, Beheshti University E-injective ⇒E -quasi injective E-injective 6⇐E -quasi injective Introduction

Preliminaries

E-quasi injectivity in Pos-S

E-quasi injectivity and θ-extensions

Reversiblity and E-quasi injectivity of S-posets

References

16/50 Some results about an E-quasi injective S-poset

Quasi injectivity of partially ordered acts Theorem Mahdieh Yavari Shahid In the category Pos-S, Beheshti University E-injective ⇒E -quasi injective E-injective 6⇐E -quasi injective Introduction

Preliminaries

E-quasi injectivity in Pos-S

E-quasi Theorem injectivity and θ-extensions In the category Pos-S,

Reversiblity Complete posets with identity actions ⇒E -quasi injective and E-quasi injectivity of S-posets

References

16/50 s e t S (left zero) r r r > @ r @ A a1 a2 @ a3 @ r@ r r @ ⊥r

Some results about an E-quasi injective S-poset

Quasi injectivity of Theorem partially ordered acts Complete poset 6⇒E -quasi injective Mahdieh Yavari Shahid Beheshti University

Introduction

Preliminaries

E-quasi injectivity in Pos-S

E-quasi injectivity and θ-extensions

Reversiblity and E-quasi injectivity of S-posets

References

17/50 > @ r @ A a1 a2 @ a3 @ r@ r r @ ⊥r

Some results about an E-quasi injective S-poset

Quasi injectivity of Theorem partially ordered acts Complete poset 6⇒E -quasi injective Mahdieh Yavari Shahid Beheshti University s e t

Introduction S (left zero) Preliminaries r r r E-quasi injectivity in Pos-S

E-quasi injectivity and θ-extensions

Reversiblity and E-quasi injectivity of S-posets

References

17/50 Some results about an E-quasi injective S-poset

Quasi injectivity of Theorem partially ordered acts Complete poset 6⇒E -quasi injective Mahdieh Yavari Shahid Beheshti University s e t

Introduction S (left zero) Preliminaries r r r E-quasi > injectivity in Pos-S @ r @ E-quasi injectivity and A a1 a2 @ a3 θ-extensions @ Reversiblity r@ r r and E-quasi @ injectivity of S-posets ⊥r References

17/50 Some results about an E-quasi injective S-poset

Quasi injectivity of partially ordered acts

Mahdieh This poset A with the action Yavari Shahid Beheshti University s t e Introduction ⊥ ⊥ ⊥ ⊥ Preliminaries a1 a1 a1 a1 E-quasi injectivity in a2 a1 a3 a2 Pos-S a3 a3 a3 a3 E-quasi injectivity and > > > > θ-extensions

Reversiblity is not E-quasi injective. and E-quasi injectivity of S-posets

References

18/50 a a0 @ r@ r @ ⊥r

Some results about an E-quasi injective S-poset

Quasi injectivity of partially ordered acts Theorem Mahdieh Yavari E-quasi injective 6⇒ Complete poset Shahid Beheshti University

Introduction

Preliminaries

E-quasi injectivity in Pos-S

E-quasi injectivity and θ-extensions

Reversiblity and E-quasi injectivity of S-posets

References

19/50 Some results about an E-quasi injective S-poset

Quasi injectivity of partially ordered acts Theorem Mahdieh Yavari E-quasi injective 6⇒ Complete poset Shahid Beheshti University

Introduction

Preliminaries a a0 E-quasi @ injectivity in r@ r Pos-S @ E-quasi injectivity and ⊥r θ-extensions

Reversiblity and E-quasi A injectivity of S-posets

References

19/50 s e t S (right zero) r r r > @ r @ A a1 @ a2 @ @ rr @ ⊥ r

Some results about an E-quasi injective S-poset

Quasi injectivity of Theorem partially ordered acts The action on an E-quasi injective S-poset need not be identity. Mahdieh Yavari Shahid Beheshti University

Introduction

Preliminaries

E-quasi injectivity in Pos-S

E-quasi injectivity and θ-extensions

Reversiblity and E-quasi injectivity of S-posets

References

20/50 > @ r @ A a1 @ a2 @ @ rr @ ⊥ r

Some results about an E-quasi injective S-poset

Quasi injectivity of Theorem partially ordered acts The action on an E-quasi injective S-poset need not be identity. Mahdieh Yavari Shahid Beheshti s e t University S (right zero)

Introduction r r r Preliminaries

E-quasi injectivity in Pos-S

E-quasi injectivity and θ-extensions

Reversiblity and E-quasi injectivity of S-posets

References

20/50 Some results about an E-quasi injective S-poset

Quasi injectivity of Theorem partially ordered acts The action on an E-quasi injective S-poset need not be identity. Mahdieh Yavari Shahid Beheshti s e t University S (right zero)

Introduction r r r Preliminaries > E-quasi @ injectivity in r @ Pos-S A a1 @ a2 E-quasi @ injectivity and θ-extensions @ rr @ Reversiblity and E-quasi ⊥ injectivity of r S-posets

References

20/50 Some results about an E-quasi injective S-poset

Quasi injectivity of partially ordered acts

Mahdieh Yavari Shahid Beheshti University Then A(S) = {f : S → A : f is a monotone map} with Introduction • pointwise order, and Preliminaries • the action given by (fs)(t) = f (st), for s, t ∈ S and f ∈ A(S) E-quasi injectivity in is E-quasi injective and has non zero elements. Pos-S

E-quasi injectivity and θ-extensions

Reversiblity and E-quasi injectivity of S-posets

References

21/50 a3 a4 rr a1 a2 ,→ a1 a2 rBA r r r

f : B → A, f (a1) = a3, f (a2) = a4. There does not exist an S-poset map f¯ : A → A which extends f .

Some results about an E-quasi injective S-poset

Quasi injectivity of partially Proposition ordered acts

Mahdieh There exists no po-monoid S over which all S-posets are Yavari Shahid E-quasi injective. Beheshti University

Introduction

Preliminaries

E-quasi injectivity in Pos-S

E-quasi injectivity and θ-extensions

Reversiblity and E-quasi injectivity of S-posets

References

22/50 Some results about an E-quasi injective S-poset

Quasi injectivity of partially Proposition ordered acts

Mahdieh There exists no po-monoid S over which all S-posets are Yavari Shahid E-quasi injective. Beheshti University a3 a4 Introduction rr Preliminaries

E-quasi a1 a2 ,→ a1 a2 injectivity in Pos-S rBA r r r E-quasi injectivity and θ-extensions f : B → A, f (a ) = a , f (a ) = a . Reversiblity 1 3 2 4 and E-quasi There does not exist an S-poset map f¯ : A → A which extends injectivity of S-posets f . References

22/50 But the converse is not true in general. Remark The product of E-quasi injective S-posets, is not necessarily E-quasi injective.

Behaviour of E-quasi injective S-posets with direct products

Quasi injectivity of partially ordered acts Theorem Mahdieh Q Yavari Let {Ai : i ∈ I } be a family of S-posets. If the product i∈I Ai Shahid Beheshti is E-quasi injective in the category Pos-S and has a zero University element θ = (θi )i∈I then each Ai is E-quasi injective in Pos-S.

Introduction

Preliminaries

E-quasi injectivity in Pos-S

E-quasi injectivity and θ-extensions

Reversiblity and E-quasi injectivity of S-posets

References

23/50 Behaviour of E-quasi injective S-posets with direct products

Introduction

Preliminaries

E-quasi injectivity in Pos-S But the converse is not true in general.

E-quasi injectivity and Remark θ-extensions The product of E-quasi injective S-posets, is not necessarily Reversiblity and E-quasi E-quasi injective. injectivity of S-posets

References

23/50 Relation between E-quasi injectivity and E-injectivity

Quasi injectivity of partially ordered acts

Mahdieh Yavari Shahid Beheshti University Theorem

Introduction An E-quasi injective S-poset A is E-injective if and only if A (S) Preliminaries has a zero element and A × A¯ is E-quasi injective (A¯ is the E-quasi Dedekind-MacNeille completion of A). injectivity in Pos-S

E-quasi injectivity and θ-extensions

Reversiblity and E-quasi injectivity of S-posets

References

24/50 s e t S (left zero) r r r

a1 a2 a3 a4 A r r r r

A, E-quasi injective 6⇒ A ⊕ {θ}, E-quasi injective

Quasi injectivity of Deﬁnition partially ordered acts An S-poset A ⊕ {θ}, which is obtained by adjoining a zero top Mahdieh element θ to an S-poset A, is called the θ-extension of A. Yavari Shahid Beheshti University

Introduction

Preliminaries

E-quasi injectivity in Pos-S

E-quasi injectivity and θ-extensions

Reversiblity and E-quasi injectivity of S-posets

References

25/50 a1 a2 a3 a4 A r r r r

A, E-quasi injective 6⇒ A ⊕ {θ}, E-quasi injective

Introduction s e t Preliminaries S (left zero)

E-quasi injectivity in r r r Pos-S

E-quasi injectivity and θ-extensions

Reversiblity and E-quasi injectivity of S-posets

References

25/50 A, E-quasi injective 6⇒ A ⊕ {θ}, E-quasi injective

Introduction s e t Preliminaries S (left zero)

E-quasi injectivity in r r r Pos-S

E-quasi injectivity and a1 a2 a3 a4 θ-extensions A Reversiblity and E-quasi injectivity of r r r r S-posets

References

25/50 A, E-quasi injective 6⇒ A ⊕ {θ}, E-quasi injective

Quasi injectivity of partially ordered acts Mahdieh The poset A with the following action is an E-quasi injective Yavari Shahid S-poset. But, A ⊕ {θ} is not E-quasi injective. Beheshti University

Introduction s t e Preliminaries a1 a1 a1 a1 E-quasi injectivity in a2 a1 a3 a2 Pos-S a3 a3 a3 a3 E-quasi injectivity and a4 a3 a1 a4 θ-extensions

Reversiblity and E-quasi injectivity of S-posets

References

26/50 Kθ-quasi injectivity

Quasi injectivity of partially ordered acts Deﬁnition Mahdieh Yavari An S-poset A ⊕ {θ} is called Kθ-quasi injective if for every sub Shahid Beheshti S-poset B of A ⊕ {θ}, any S-poset map f : B → A ⊕ {θ}, with University ←− f (θ) = {b ∈ B : f (b) = θ}= 6 ∅, can be extended to Introduction f¯ : A ⊕ {θ} → A ⊕ {θ}. Preliminaries E-quasi B A ⊕ {θ} injectivity in / Pos-S ←− f , with f (θ)6=∅ E-quasi f¯ injectivity and y θ-extensions A ⊕ {θ} Reversiblity and E-quasi injectivity of S-posets

References

27/50 Example Whenever S = G is a group, every subact B of a G-act A is a G-ﬁlter.

S-ﬁlter

Quasi injectivity of partially ordered acts

Mahdieh Deﬁnition Yavari Shahid Let A be an S-act. A subset B of A is called consistent if for Beheshti University each a ∈ A and s ∈ S, as ∈ B implies a ∈ B. We call a consistent subact an S-ﬁlter. Introduction

Preliminaries

E-quasi injectivity in Pos-S

E-quasi injectivity and θ-extensions

Reversiblity and E-quasi injectivity of S-posets

References

28/50 S-ﬁlter

Quasi injectivity of partially ordered acts

Preliminaries

E-quasi injectivity in Pos-S Example E-quasi injectivity and θ-extensions Whenever S = G is a group, every subact B of a G-act A is a

Reversiblity G-ﬁlter. and E-quasi injectivity of S-posets

References

28/50 Kθ-quasi injectivity

Quasi injectivity of partially ordered acts Theorem Mahdieh Yavari Shahid Let A be an E-quasi injective S-poset. Also, assume that Beheshti f : B → A ⊕ {θ} is an S-poset map, where B is a sub S-poset University ←− of A ⊕ {θ} and f (θ) 6= ∅. Then there exists an S-ﬁlter A˜ of Introduction A ⊕ {θ} which is Preliminaries • upward closed in A ⊕ {θ}, E-quasi ←− injectivity in ˜ Pos-S • f (θ) ⊆ A, and ˜ E-quasi • A ∩ {b ∈ B : f (b) 6= θ} = ∅ injectivity and θ-extensions if and only if there exists an S-poset map ¯ Reversiblity f : A ⊕ {θ} → A ⊕ {θ} which extends f . and E-quasi injectivity of S-posets

References

29/50 Kθ-quasi injectivity

Quasi injectivity of partially ordered acts

Mahdieh Yavari Corollary Shahid Beheshti University Let A be an E-quasi injective S-poset. If for each S-poset map f : B → A ⊕ {θ}, where B is a sub S-poset of A ⊕ {θ} and ←− Introduction f (θ) 6= ∅, we have Preliminaries E-quasi ←− injectivity in A˜ = {a ∈ A ⊕ {θ} : ∃s ∈ S, as ∈↑ f (θ)} Pos-S

E-quasi injectivity and is an S-ﬁlter of A ⊕ {θ}, then A ⊕ {θ} is Kθ-quasi injective. θ-extensions

Reversiblity and E-quasi injectivity of S-posets

References

30/50 Kθ-quasi injectivity

Quasi injectivity of partially ordered acts

Mahdieh Corollary Yavari Shahid Suppose S is a po-monoid with any one of the following Beheshti University properties: (1) ∀s ∈ S, ∃t ∈ S, st ≤ e. Introduction (2) ∀s ∈ S, s2 ≤ e. Preliminaries (3) S is a po-group. E-quasi injectivity in (4) > = e (> is the top element of S). Pos-S S S (5) S is a right zero semigroup with an adjoined identity. E-quasi injectivity and If A is an E-quasi injective S-poset then A ⊕ {θ} is K -quasi θ-extensions θ

Reversiblity injective in Pos-S. and E-quasi injectivity of S-posets

References

31/50 Decreasing action

Quasi injectivity of partially ordered acts

Mahdieh Yavari Shahid Beheshti University Deﬁnition

Introduction Let A be an S-poset. Then, the action on A is called Preliminaries decreasing (contractive, non expansive) if for every a ∈ A and E-quasi s ∈ S, as ≤ a. injectivity in Pos-S

E-quasi injectivity and θ-extensions

Reversiblity and E-quasi injectivity of S-posets

References

32/50 Example for Decreasing action

∞ Quasi injectivity of partially r ordered acts p p 3 Mahdieh p Yavari r Shahid ∞ Beheshti S = = ∪ {∞} 2 University N N r Introduction 1 Preliminaries r E-quasi injectivity in • With the binary operation m.n = min{m, n}, S is a Pos-S

E-quasi po-monoid and ∞ is the identity element of S. injectivity and θ-extensions • S has the properties (1), (2), and (4) in previous corollary. ∞ Reversiblity • Also, every N -poset has the property that an ≤ a∞ = a, and E-quasi ∞ ∞ injectivity of for all n ∈ N . Therefore, the action on every N -poset is S-posets decreasing. References

33/50 Kθ-quasi injectivity

Quasi injectivity of partially ordered acts

Mahdieh Yavari Shahid Beheshti University Corollary

Introduction Let D be a full subcategory of the category Pos-S, where the

Preliminaries action on every object A ∈ D is decreasing. If A is E-quasi E-quasi injective in D then A ⊕ {θ} is Kθ-quasi injective in D. injectivity in Pos-S

E-quasi injectivity and θ-extensions

Reversiblity and E-quasi injectivity of S-posets

References

34/50 Reversible automata

Quasi injectivity of partially ordered acts

Mahdieh Yavari Shahid Revesible automata are important kinds of automata and have Beheshti University studied under various names. Glushkov in [3] called them invertible and Thierrin in [5] called them locally transitive (see Introduction also [1]). Since each S-act is a kind of automaton and because Preliminaries

E-quasi of the importance of reversible automata in automata theory, injectivity in Pos-S we introduce the category of reversible partially ordered acts

E-quasi and study the notion of E-quasi injectivity in this category. injectivity and θ-extensions

Reversiblity and E-quasi injectivity of S-posets

References

35/50 Reversible S-poset

Quasi injectivity of partially ordered acts

Mahdieh Yavari Shahid Beheshti University Deﬁnition An S-poset A is called reversible if for every a ∈ A and s ∈ S, Introduction there exists t ∈ S such that ast = a. So, we have the category Preliminaries R-Pos-S of all reversible S-posets with S-poset maps between E-quasi injectivity in them. Pos-S

E-quasi injectivity and θ-extensions

Reversiblity and E-quasi injectivity of S-posets

References

36/50 Example of Reversible S-poset

Quasi s e t injectivity of partially S(right zero) ordered acts r r r Mahdieh Yavari Shahid Beheshti a1 a2 University @ r@ r Introduction A @ Preliminaries ⊥ E-quasi r injectivity in Pos-S

E-quasi s t e injectivity and θ-extensions a1 a2 a1 a1

Reversiblity a2 a2 a1 a2 and E-quasi injectivity of ⊥ ⊥ ⊥ ⊥ S-posets

References

37/50 Strong Reversible S-poset

Quasi injectivity of partially ordered acts

Mahdieh Yavari Shahid Beheshti University Deﬁnition An S-poset A is called strong reversible if for every s ∈ S there Introduction exists t ∈ S such that ast = ats = a for all a ∈ A. So, we have Preliminaries the category SR-Pos-S of all strong reversible S-posets and E-quasi injectivity in S-poset maps between them. Pos-S

E-quasi injectivity and θ-extensions

Reversiblity and E-quasi injectivity of S-posets

References

38/50 Example of Strong Reversible S-poset

Quasi injectivity of partially ordered acts

Mahdieh Yavari Shahid s e t v Beheshti University S

Introduction r r r r Preliminaries s t v e E-quasi injectivity in s t v s s Pos-S t v s t t E-quasi injectivity and v s t v v θ-extensions e s t v e Reversiblity and E-quasi injectivity of S-posets

References

39/50 Example of Strong Reversible S-poset

Quasi injectivity of partially ordered acts

Mahdieh Yavari Shahid Beheshti a b c University A Introduction r r r Preliminaries

E-quasi s t v e injectivity in Pos-S a b c a a E-quasi b c a b b injectivity and θ-extensions c a b c c Reversiblity and E-quasi injectivity of S-posets

References

40/50 Square Reversible S-poset

Quasi injectivity of partially ordered acts

Mahdieh Yavari Shahid Beheshti University Deﬁnition

Introduction An S-poset A is square reversible if for every a ∈ A and s ∈ S, 2 Preliminaries we have as = a. So, we have the category SQ-Pos-S of all E-quasi square reversible S-posets and S-poset maps between them. injectivity in Pos-S

E-quasi injectivity and θ-extensions

Reversiblity and E-quasi injectivity of S-posets

References

41/50 Example of Suare Reversible S-poset

Quasi injectivity of partially ordered acts

Mahdieh s e t a1 a2 Yavari SA Shahid Beheshti University r r r r r

Introduction s t e Preliminaries s t s s E-quasi t s t t injectivity in Pos-S e s t e E-quasi injectivity and θ-extensions

Reversiblity s t e and E-quasi injectivity of a1 a2 a1 a1 S-posets a2 a1 a2 a2 References

42/50 Relation between square, strong and reversible

Quasi injectivity of partially ordered acts

Mahdieh Yavari Shahid Beheshti University Square reversible ⇒ Strong reversible ⇒ Reversible

Introduction

Preliminaries • But the converses are not true in general. E-quasi • S po-group ⇒ every S-poset is strong reversible. injectivity in Pos-S • S as an S-poset is strong reversible ⇐⇒ S is a group.

E-quasi injectivity and θ-extensions

Reversiblity and E-quasi injectivity of S-posets

References

43/50 Kθ-quasi injectivity

Quasi injectivity of partially ordered acts

Mahdieh Yavari Shahid Beheshti University Corollary

Introduction Let A be an E-quasi injective (strong, square) reversible Preliminaries S-poset. Then A ⊕ {θ} is Kθ-quasi injective in R-Pos-S E-quasi (SR-Pos-S, SQ-Pos-S). injectivity in Pos-S

E-quasi injectivity and θ-extensions

Reversiblity and E-quasi injectivity of S-posets

References

44/50 A, Kθ-quasi injective 6⇒ A \{θ}, E-quasi injective

Quasi injectivity of θ partially ordered acts @ Mahdieh r @ Yavari Shahid a4 @ a5 Beheshti @ University r@ r @ a3 Introduction @ Preliminaries r @ a @ a E-quasi 1 2 injectivity in @ Pos-S @ rr

E-quasi @ injectivity and ⊥ θ-extensions r Reversiblity and E-quasi injectivity of S-posets

References

45/50 Corollary (i) In SR-Pos-S, an object A is E-quasi injective if it is complete. (ii) In SQ-Pos-S, an object A is E-quasi injective if it is complete.

Relation between E-quasi injectivity and completeness

Quasi injectivity of partially ordered acts Theorem Mahdieh Let A be a strong (square) reversible S-poset. Then the Yavari Shahid following are equivalent in the category SR-Pos-S (SQ-Pos-S): Beheshti University (i) A is E-injective. (ii) A is E-absolute retract. Introduction (ii) A is complete. Preliminaries

E-quasi injectivity in Pos-S

E-quasi injectivity and θ-extensions

Reversiblity and E-quasi injectivity of S-posets

References

46/50 Relation between E-quasi injectivity and completeness

E-quasi injectivity in Pos-S Corollary E-quasi (i) In SR-Pos-S, an object A is E-quasi injective if it is injectivity and θ-extensions complete. Reversiblity (ii) In SQ-Pos-S, an object A is E-quasi injective if it is and E-quasi injectivity of complete. S-posets

References

46/50 References

Quasi injectivity of partially ordered acts Ahsan, J.: Monoids characterized by their quasi injective Mahdieh Yavari S-systems, emigroup Forum 36 (1987), 285-292. Shahid Beheshti University Banaschewski, B.: Injectivity and essential extensions in equational classes of algebras, Queen’s Papers in Pure Introduction Appl. Math. 25 (1970), 131-147. Preliminaries E-quasi Bulman-Fleming, S. and M. Mahmoudi: The category of injectivity in Pos-S S-posets, Semigroup Forum 71(3) (2005), 443-461. E-quasi injectivity and Ebrahimi, M.M.: Internal completeness and injectivity of θ-extensions Boolean algebras in the topos of M-set, Bull. Austral. Reversiblity and E-quasi Math. 41(2) (1990), 323-332. injectivity of S-posets

References

47/50 References

Quasi injectivity of Ehrig, H., F. Parisi-Presicce, P. Bohem, C. Rieckhoff, C. partially ordered acts Dimitrovici, and M. Grosse-Rhode: Combining data type Mahdieh and recursive process specifications using projection Yavari Shahid algebras, Theoret. Comput. Sci. 71 (1990), 347-380. Beheshti University Fakhruddin, S.M.: Absolute flatness and amalgams in Introduction pomonoids, Semigroup Forum 33 (1986), 15-22. Preliminaries Glushkove, V.M., Abstraktnaya teoriya avtomatov, Uspekhi E-quasi injectivity in Matem. Nauk 16(5) (1961), 3-62 [English translation: The Pos-S abstract theory of automata, Russ. Math. Surveys 16(5) E-quasi injectivity and (1961),1-53]. θ-extensions Reversiblity Guili, E.: On m-separated projection spaces, Appl. and E-quasi injectivity of Categorical structures 2 (1994), 91-99. S-posets References Kilp, M., U. Knauer, and A. Mikhalev: “Monoids, Acts and

Categories”, Walter de Gruyter, Berlin, New York, 2000. 48/50 References

Quasi injectivity of Kovaevi, J., M. iri, T. Petkovi, and S. Bogdanovi, partially ordered acts Decompositions of automata and reversible states, Mahdieh Publicationes Mathematicae Debrecen 60 (3-4) (2002), Yavari Shahid 587-602. Beheshti University Lopez Jr. A.M., and J.K. Luedeman: Quasi injective Introduction S-systems and their S-endomorphism semigroups, Preliminaries Czechoslovak Math. J. 29(104) (1979), 97-104. E-quasi injectivity in Roueentan, M., M. Ershad: Strongly duo and duo right Pos-S S-acts, Italian J. Pure Appl. Math 32 (2014) 143-154. E-quasi injectivity and θ-extensions Sikorski, R.: “Boolean Algebras”, 3rd edn. Springer, New Reversiblity York 1969. and E-quasi injectivity of S-posets Thierrin, G., Decompositions of locally transitive References semiautomata, Utilitas Mathematica 2 (1972), 25-32.

49/50 Quasi injectivity of partially ordered acts Mahdieh Thank You For Your Attention Yavari Shahid Beheshti University

Introduction

Preliminaries

E-quasi injectivity in Pos-S

E-quasi injectivity and θ-extensions Shahid Beheshti University Reversiblity and E-quasi injectivity of S-posets

References

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