Injective hulls for S-posets Xia Zhang South China Normal...
Transcript of 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
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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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3. Injective hulls for S-posets Definitions
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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3. Injective hulls for S-posets Definitions
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
Theorem 10 (2015, Zhang, Laan)
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.
Theorem 12 (2016, Zhang, Laan)
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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4. Injective hulls for ordered algebras Constructions of
injective hulls in OALg≤Ω
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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4. Injective hulls for ordered algebras Constructions of
injective hulls in OALg≤Ω
Step 3
For any ordered Ω-algebra A = (A,ΩA,≤A), we denote
Q(A) := P(A)cl = D ∈P(A) | cl(D) = D.
Theorem 13 (2016, Zhang, Laan)
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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