injective hulls for s-posets xia zhang south china normal...

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Injective hulls for S-posets

Xia Zhang

South China Normal University

Joint work with Valdis Laan

International Workshop on Graphs, Semigroups, and Semigroup Acts

Berlin, October 11 2017

Xia Zhang, Valdis Laan (SCNU) 10. 11. 2017 0 / 39

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Contents

1 Introduction1 Backgrounds: injective hulls for posets2 Backgrounds: injective hulls for semilattices3 Backgrounds: injective hulls for certain S-systems over semilattices4 Backgrounds: injective hulls for po-monoids5 Basic definitions

2 Injective hulls of posemigroups1 Injective objects in Pos≤2 Construction of a special closure operator3 Injective hulls for posemigroups

3 Injective hulls for S-posets1 Injective objects in Pos≤S2 Construction of a special closure operator3 Injective hulls for S-posets

4 Injective hulls for ordered algebras5 Work to be continued6 References


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1. Introduction: Results of injective hulls on posets

1967, Banaschewski B., Bruns G., Categorical construction of theMacNeille completion, Arch. Math.

Theorem 1 (Banaschewski B., Bruns G.) For a partially ordered set P,T.A.E.:

1 P is complete;

2 P is strictly injective;

3 P is a retract of every extension;

4 P has no essential extensions.


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1. Introduction: Results of injective hulls on posets

Theorem 2(Banaschewski B., Bruns G.) For a partially ordered set P, T.A.E.:

1 E is a MacNeille completion of P;

2 E is an essential, strictly injective extension of P;

3 E is a strictly injective extension of P not containing any properly smallersuch extension of P;

4 E is an essential extension of P not containing in any properly larger suchextension of P.


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1. Introduction: Results of injective hulls on semilattices

1970, Burns G., Lakser H., injective hulls of semilattices. Bulletin

Theorem 3 (Burns G., Lakser H.)A semilattice S is injective iff it is complete and satisfying

a∧∨

M =∨(

a∧

m | m ∈ M), (1)

for all a ∈ S , M ⊆ S .


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1. Introduction: Results of injective hulls on semilattices

1970, Burns G., Lakser H., injective hulls of semilattices. Bulletin

ID(S) = A ⊆ S | A = A ↓;

M ⊆ A with∨

M exists and satisfying (1)⇒∨

M ∈ A.

Theorem 4 (Burns G., Lakser H.)

Injective hulls of a semilattice S is ID(S).


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1. Introduction: Results of injective hulls on certainS-systems over a semilattice

1972, Johnson C.S., J.R., McMorris F.R., injective hulls oncertain S-systems over a semilattice. Proc. Amer. Math. Soc.

Theorem 5 (Johnson C.S., J.R., McMorris F.R.)

Injective hulls of an S-system MS is ID(MS).


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1. Introduction: backgrounds and motivations

1974, Schein B.M. Injectives in certain classes of semigroups.Semigroup Forum.

Theorem 6. (1974 Schein)The category of semigroups has only trivialinjectives.


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1. Introduction: Results of injective hulls on posemigroups

2012, Lambek J., Barr M., Kennison J.F. and Raphael R.,Injective hulls of partially ordered monoids. Theory Appl. Categ.

The category of po-monoids

Partially ordered monoids (po-monoid) with submultiplicativeorder-preserving mappings, i.e., an order-preserving mapping φ : A→ B ofpo-monoids satisfying

φ(1) = 1,

φ(a)φ(b) ≤ φ(ab),

for all a, b ∈ A.


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1. Introduction: Results of injective hulls on posemigroups

2012, Lambek J., Barr M., Kennison J.F. and Raphael R.,Injective hulls of partially ordered monoids. Theory Appl. Categ.

Theorem 7 (2012 Lambek, Barr, Kennison and Raphael)A po-monoid is injective iff it is a quantale.


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1. Introduction: Quantales and quantale-like structuresDefinitions

Definition 1 (Mulvey C.J., 1986)

A quantale is a complete lattice Q equipped with an associative binary operationsatisfying

a(∨

M) =∨am | m ∈ M, (

∨M)a =

∨ma | m ∈ M

for any a ∈ Q, M ⊆ Q.

A frame(locale) L is a complete lattice such that

a∧

(∨

M) =∨a∧

m | m ∈ M,

for any a ∈ L, M ⊆ L.


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1. Introduction: Quantale modules Definitions

Definition 2 (Cf. [5, 9])

Given a quantale Q, a left quantale module is a pair (A, ∗), where A is a∨

-lattice

and Q × A∗→ such that:

1 q ∗ (∨

S) =∨

s∈S(q ∗ s) for every q ∈ Q, S ⊆ A;

2 (∨

T ) ∗ a =∨

t∈T (t ∗ a) for every a ∈ A, T ⊆ Q;

3 q1 ∗ (q2 ∗ a) = (q1q2) ∗ a for every a ∈ A, q1, q2 ∈ Q.


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1. Introduction: Quatale algebras Definitions

Definition 3 (Cf. [9]) Given a commutative quantale Q, a quantalealgebra is a quatnale module (A, ∗) such that:

1 (A,≤,⊗) is a quantale;

2 q ∗ (a⊗ b) = (q ∗ a)⊗ b for every a, b ∈ A, q ∈ Q.


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2. Injective hulls for posemigroups Category Pos≤

Pos≤Objects: posemigroups;

Morphisms: order-preserving submultiplicative mappings;

ε≤ : a morphism f : S → T in Pos≤ belongs to ε≤ if it satisfies:

f (a1) · · · f (an) ≤ f (a) =⇒ a1 · · · an ≤ a,

for each a1, · · · , an, a ∈ S .


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2. Injective hulls for posemigroups Injectivies

Theorem 8 (2014, Zhang, Laan)

ε≤-injective objects in the category Pos≤ are exactly quantales.

Example: quantale (P(S), ·,⊆)

S : a posemigroup,P(S): the set of all downsets of S ,

X · Y = (XY ) ↓= x ∈ S | s ≤ xy for some x ∈ X , y ∈ Y ,

for X ,Y ∈P(S).


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2. Injective hulls for posemigroups Nucleus

Definition 4: Nucleus

a nucleus on a quantale Q is a submultiplicative closure operator on Q.

the nucleus cl:

For any down-set I of a posemigroup S we define its closure by

cl(I ) := x ∈ S | aIc ⊆ b ↓ implies axc ≤ b for all a, b, c ∈ S.

Then cl : P(S)→P(S) is a nucleus.


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2. Injective hulls for posemigroups Constructions

Construction: Q(S)

Q(S) := I ∈P(S) | I = cl(I )

define a multiplication on Q(S) by

I J := cl(I · J). (1)

Theorem 9 (2014 Zhang, Laan)[Cf. Theorem 5.8 in [6]]

Let S be a posemigroup such that cl(s ↓) = s ↓ for every s ∈ S . ThenQ(S) is an ε≤-injective hull of S in Pos≤.


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3. Injective hulls for S-posets Definitions

In this work, S is always a pomonoid.

S-posets

A poset (A, ≤) together with a mapping A× S → A (under which a pair (a, s)maps to an element of A denoted by as) is called a right S-poset, denoted by AS ,if for any a, b ∈ A, s, t ∈ S ,

1 a(st) = (as)t,

2 a1 = a,

3 a ≤ b, s ≤ t imply that as ≤ bt.

A left S-poset can be defined similarly.


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3. Injective hulls for S-posets Backgrounds

S : a pomonoid

Fakhruddin S.M., On the category of S-posets. Acta Sci. Math., 1988

Result 1:Each injective S-poset is “complete” if it is a complete lattice and satisfying the

distributions of arbitrary joins with S-actions.

Result 2:Let G be a pogroup. Then a comlete G -poset is injective.


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Definition 5 (2015, Zhang, Laan)

We call a right S-poset AS a right S-quantale if

1 the poset A is a complete lattice;

2 (∨

M)s =∨ms | m ∈ M for each subset M of A and each s ∈ S .


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Let AS and BS be S-posets. We say that a mapping f : A→ B isS-submultiplicative if f (a)s ≤ f (as) for any a ∈ A, s ∈ S .

Pos≤S

We denote by Pos≤S the category where objects are right S-posets and morphisms

are S-submultiplicative order-preserving mappings.

ε≤

Let ε≤ be the class of morphisms e : AS → BS in the category Pos≤S which satisfythe following condition:

e(a)s ≤ e(a′) =⇒ as ≤ a′, ∀ a, a′ ∈ A, s ∈ S .


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3. Injective hulls for S-posets Injective objects


Let AS be an S-poset. Then AS is ε≤-injective in Pos≤S if and only if AS is a

right S-quantale.

An S-quantale

Let AS be an S-poset, and let P(A) be the set of all down-sets of theposet A. Define a right S-action · on P(A) by

D · s = (Ds) ↓= x ∈ A | x ≤ ds for some d ∈ D,

for any s ∈ S , D ∈P(A). Then (P(A), ·,⊆) is a right S-quantale.


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3. Injective hulls for S-posets Further constructions

A nucleusFor any down-set D of an S-poset AS we define its closure by

cl(D) := x ∈ A | Ds ⊆ a ↓ implies xs ≤ a for all a ∈ A, s ∈ S.

Then cl is a nucleus on P(A)S .


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3. Injective hulls for S-posets Further constructions

Another S-quantale

For any S-poset AS , we put

Q(A) := P(A)cl = D ∈P(A) | cl(D) = D

and define a right S-action on Q(A) by

D s := cl(D · s),

for any s ∈ S .


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3. Injective hulls for S-posets Further constructions: injective

hulls

Theorem 11 (2015 Zhang, Laan)

For every S-poset AS , Q(A)S is the ε≤-injective hull of AS in the categoryPos≤S .


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4. Injective hulls for ordered algebras Definitions

Definition 6

Let Ω be a type. An ordered Ω-algebra is a triplet A = (A,ΩA,≤A) comprising aposet (A,≤A) and a set ΩA of operations on A (for every k-ary operation symbolω ∈ Ωk there is a k-ary operation ωA ∈ ΩA on A) such that all the operations ωA

are monotone mappings, where monotonicity of ωA (ω ∈ Ωk) means that

a1 ≤A a′1 ∧ . . . ∧ ak ≤A a′k =⇒ ωA(a1, . . . , ak) ≤A ωA(a′1, . . . , a′k)

for all a1, . . . , ak , a′1, . . . , a

′k ∈ A.


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4. Injective hulls for ordered algebras Definition:

sup-algebras

An ordered Ω-algebra A = (A,ΩA,≤A) is called a sup-algebra if theposet (A,≤) is a complete lattice and

ωA(a1, . . . , ai−1,∨

M, ai+1, . . . , an)

=∨ωA(a1, . . . , ai−1,m, ai+1, . . . , an) | m ∈ M

for every n ∈ N, ω ∈ Ωn, i ∈ 1, . . . , n, a1, . . . , ai−1, ai+1, . . . , an ∈ A,and M ⊆ A.

Sup-algebras are a generalization of quantale-like structures.


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4. Injective hulls for ordered algebras Definition:

subhomomorphism

Definition 7Let A ,B be ordered Ω-algebras of type Ω. A monotone mapping f : A→ B iscalled a subhomomorphism if

ωB(f (a1), . . . , f (an)) ≤ f (ωA(a1, . . . , an))

for every n ∈ N, ω ∈ Ωn, a1, . . . , an ∈ A, and

ωB ≤ f (ωA)

for every ω ∈ Ω0. All ordered Ω-algebras together with their subhomomorphisms

form a category which we denote by OALg≤Ω .


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4. Injective hulls for ordered algebras Injectives

We will study injectivity in classes of ordered Ω-algebras with respect totwo classes of order-embeddings. The first of them is the class M1 oforder-embeddings that are homomorphisms. The other class has a morecomplicated definition.

Let us denote by T nΩ the set of all n-ary Ω-terms. Let M2 be the class of

mappings h : A → B between ordered Ω-algebras that satisfy thefollowing conditions:

1 h is monotone,

2 ωB(h(a1), . . . , h(an)) ≤ h(ωA(a1, . . . , an)) for every n ∈ N, ω ∈ Ωn,a1, . . . , an ∈ A,

3 h(ωA) = ωB for all ω ∈ Ω0,

4 for all n ∈ N, t ∈ T nΩ, a1, . . . , an, a ∈ A,

tB(h(a1), . . . , h(an)) ≤ h(a) =⇒ tA(a1, . . . , an) ≤ a.


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4. Injective hulls for ordered algebras Injectives

Lemma 2For a class of ordered Ω-algebras, M1 ⊆M2.


Every sup-algebra Q = (Q,ΩQ ,≤Q) of type Ω is M2-injective and therefore,

M1-injective in the category OALg≤Ω .


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Proof.Let h : A→ B be a morphism in M2 and let f : A→ Q be a morphism inOALg≤

Ω . Define a mapping g : B → Q by

g(b) =∨t(f (a1), . . . , f (am)) |

m ∈ N, a1, . . . , am ∈ A, t ∈ TmΩ , t(h(a1), . . . , h(am)) ≤ b,

for any b ∈ B. We write the last join shortly as∨t(h(a1),...,h(am))≤b

t(f (a1), . . . , f (am)).


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ωQ(g(b1), . . . , g(bn))

= ωQ

(∨t1(f (a11), . . . , f (a1m1))

∣∣ t1(h(a11), . . . , h(a1m1)) ≤ b1

, . . . ,∨

tn(f (an1), . . . , f (anmn)) | tn(h(an1), . . . , h(anmn)) ≤ bn

)=∨

ωQ (t1(f (a11), . . . , f (a1m1)), . . . , tn(f (an1), . . . , f (anmn)))∣∣

t1(h(a11), . . . , h(a1m1)) ≤ b1, . . . , tn(h(an1), . . . , h(anmn)) ≤ bn

≤∨

t(f (a1), . . . , f (am))∣∣ t(h(a1), . . . , h(am)) ≤ ωB(b1, . . . , bn)

= g(ωB(b1, . . . , bn)),


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Theorem 6(2016, Zhang, Laan[12])

For an ordered Ω-algebra A = (A,ΩA,≤A), the following assertions areequivalent:

1 A is M2-injective in OALg≤Ω ,

2 A is M1-injective in OALg≤Ω ,

3 A is a sup-algebra.


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4. Injective hulls for ordered algebras Constructions of

injective hulls in OALg≤Ω

Step 1

Let A = (A,ΩA,≤A) be an ordered Ω-algebra, and let P(A) be the set of alldown-sets of the poset (A,≤A). For any n ∈ N, ω ∈ Ωn and D1, . . . ,Dn ∈P(A)we denote

ωA(D1, . . . ,Dn) = ωA(d1, . . . , dn) | d1 ∈ D1, . . . , dn ∈ Dn

and define an operation ωP(A) on P(A) by

ωP(A)(D1, . . . ,Dn) := ωA(D1, . . . ,Dn) ↓ .

For each ω ∈ Ω0 we putωP(A) := ωA ↓ .

Then that (P(A),ΩP(A),⊆) is a sup-algebra.


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Let A = (A,ΩA,≤A) be an ordered Ω-algebra. We denote by P1A the set

of all unary polynomial functions on A . Recall that a unary polynomialfunction on A is a mapping p : A→ A such that p = tA (a1, . . . , an−1,− )for some n ∈ N and a1, . . . , an−1 ∈ A. Clearly, every unary polynomialfunction is monotone.

Step 2

For any D ∈P(A) denote

cl(D) := u ∈ A | p(D) ⊆ a ↓⇒ p(u) ≤ a for all a ∈ A, p ∈ P1A,

where p(D) = p(d) | d ∈ D.


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Step 3

For any ordered Ω-algebra A = (A,ΩA,≤A), we denote

Q(A) := P(A)cl = D ∈P(A) | cl(D) = D.


Let A = (A,ΩA,≤A) be an ordered Ω-algebra such that cl(a ↓) = a ↓ for every

a ∈ A. Then Q(A) is the M2-injective hull of A in the category OALg≤Ω .


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5. Work to be continued

2017, Xia C.C., Zhao B., Han S.W., A Note on injective hulls of

posemigropus. Theory Appl. Categ.

Let (S , ·) be a posemigroup, D ⊆ S . Define

D? , Dul ∩ DL ∩ DR ∩ DT ,

whereDul , x ∈ A | (∀b ∈ A) D ⊆ b ↓=⇒ x ≤ b,

DL , x ∈ A | (∀a, b ∈ A) Da ⊆ b ↓=⇒ xa ≤ b,DR , x ∈ A | (∀a, b ∈ A) aD ⊆ b ↓=⇒ ax ≤ b,

DT , x ∈ A | (∀a, b, c ∈ A) aDc ⊆ b ↓=⇒ axc ≤ b,

S ?=D ⊆ S | D? = D is the ε≤-injective hull for a posemigroup S .


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References

[1] Adamek J., Herrlich H. and Strecker G. E., Abstract and concrete categories: Thejoy of cats, John Wiley and Sons, New York, 1990.

[2] Banaschewski B., Bruns G., Categorical construction of the MacNeille completion.Arch. Math., 1967, 369-377.[3] Bruns, G. and Lakser, H. Injective hulls of semilattices. Canad. Math. Bull., 1970,13, 115–118.

[4]Fakhruddin S.M., On the category of S-posets. Acta Sci. Math., 1988, 52, 85–92.

[5] Johnson C.S., J.R., McMorris F.R., Injective hulls on certain S-systems over asemilattice. Proc. Amer. Math. Soc., 1972, 32, 371-375. [6] Kruml, D., Paseka, J.:

Algebraic and categorical aspects of quantales. In: Handbook of Algebra, vol. 5, pp.323–362. Elsevier (2008) [7] Lambek J., Barr M., Kennison J.F. and Raphael R.

Injective hulls of partially ordered monoids. Theory Appl. Categ., 2012, 26, 338–348

[8] Rosenthal K.I. Quantales and their applications. Pitman Research Notes in

Mathematics 234, Harlow, Essex, 1990.


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References

[9] Rasouli, H, Completion of S-posets, Semigroup Forum, 85 (2012), 571-?76. [10]

Schein B.M. Injectives in certain classes of semigroups. Semigroup Forum, 1974, 9,159–171.

[11] Solovyov S., A representation theorem for quantale algebras. Contr. Gen. Alg.,2008, 18, 189–198.

[12] Xia C.C., Zhao B., Han S.W., A Note on injective hulls of posemigropus. TheoryAppl. Categ. 2017, 32, 254-257.

[13] Zhang X., Laan V. On injective hulls of S-posets. Semigroup Forum, 2015, 91,62-70

[14] Zhang X. Laan V. Injective hulls for posemigroups. Proc. Est. Acad. Sci. 2014, 63,372-378.

[15] Zhang X. Laan V. Injective hulls for ordered algebras. Algebra Universalis 2016, 76,

339-349.


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Thank you!


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