On injective constructions of S -semigroups Jan Paseka Masaryk - - PowerPoint PPT Presentation

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On injective constructions of S-semigroups Jan Paseka Masaryk University

Joint work with Xia Zhang South China Normal University

BLAST 2018 University of Denver, Denver, USA

Jan Paseka (MU)

• 10. 8. 2018

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Contents

1 Introduction

M Motivation S Sierpi´ nski space and injectivity B Complete Boolean algebras and injectivity Q Backgrounds: Quantales and quantale-like structures H Backgrounds: Injective hulls for some partially ordered algebras

2 Injective hulls of posemigroups

O Injective objects of posemigroups C Construction of a special closure operator H Injective hulls of posemigroups

3 Injective hulls for S-semigroups

O Injective objects of S-semigroups C Construction of a special closure operator H Injective hulls of S-semigroups

4 References Jan Paseka (MU)

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Introduction: Motivation

Fuzzy logic

It is well-known that the semantics of a given fuzzy logic can be formally axiomatized by means of a residuated poset. The appropriateness of such approach is emphasized by the fact that t-norm based logic usually refers to residuated systems of fuzzy logic with t-norm based semantics. Here the conjunction connective is interpreted by a t-norm and the implication

• perator by its residuum. In essence, this fact is the source of numerous

examples of prospective truth functions of connectives for fuzzy logic. Hence, we assume that a corresponding residuated poset is given and its connection to the semantics of a fuzzy logic is known.

Content of the work

In this paper, we shall investigate the injective hulls in the category of S-semigroups over a posemigroup S and we obtain their concrete form.

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• 1. Introduction: Sierpi´

nski space and injectivity

We shall write S for the Sierpinski space, i.e., the set 2 = {0, 1} equipped with topology Ω(S) = {S, {1}, ∅}.

Statement

(i) Any T0-space may be embedded as a subspace of a power of S. (ii) S is an injective T0-space, i.e., if i : X ′ → X is an inclusion of T0-spaces, then every continuous map f : X ′ → S is the restriction of some continuous map g : X → S. Hence, g ◦ i = f . X ′✲ i

✲ X

S

✛

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . g f

✲

(iii) A T0-space is injective iff it is a retract of a power of S.

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• 1. Introduction: Complete Boolean algebras and injectivity

Statement

(i) The injective objects in the category Bool of Boolean algebras are precisely the complete Boolean algebras. (ii) The injective objects in the category DLat of distributive lattices are precisely the complete Boolean algebras. C✲ i

✲ D

B

✛

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . g f

✲

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• 1. Introduction: Quantales and quantale-like structures

Definitions

Definition 1 (Mulvey C.J., 1986)

A structure (Q, , ⊗) is called a quantale if (Q, ) is a -semilattice, (Q, ⊗) is a semigroup, and multiplication distributes over arbitrary joins in both coordinates, that is, a ⊗

• M
• =
• {a ⊗ m | m ∈ M},
• M
• ⊗ a =
• {m ⊗ a | m ∈ M},

for any a ∈ Q, M ⊆ Q. A quantale is commutative if the binary operation is commutative.

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• 1. Introduction: Quantales and quantale-like structures

Definitions Each quantale Q is residuated in the following natural way:

s ⊗ a ≤ b ⇐ ⇒ s ≤ a − →r b ⇐ ⇒ a ≤ s − →l b, ∀a, b, s ∈ Q.

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• 1. Introduction: Quantales and quantale-like structures

Definitions 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. [6, 11])

Given a quantale Q, a left quantale module is a pair (A, ∗), where A is a -lattice and ∗: Q × A → 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) = (q1 ⊗ q2) ∗ a for every a ∈ A, q1, q2 ∈ Q.

Q-modules are equivalent to so called Q-sup-lattices, where Q is a unital (commutative) quantale. This fact was first pointed out by Stubbe in 2006 and later proved in quantale-like setting by Solovyov in 2016. Thus, Q-modules can be seen as a fuzzification of complete lattices.

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• 1. Introduction: Quantale modules

Definitions

Definition 2 (Cf. [6, 11])

Given a quantale Q, a left quantale module is a pair (A, ∗), where A is a -lattice and ∗: Q × A → 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) = (q1 ⊗ q2) ∗ a for every a ∈ A, q1, q2 ∈ Q.

Q-modules are equivalent to so called Q-sup-lattices, where Q is a unital (commutative) quantale. This fact was first pointed out by Stubbe in 2006 and later proved in quantale-like setting by Solovyov in 2016. Thus, Q-modules can be seen as a fuzzification of complete lattices.

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• 1. Introduction: Quantale algebras

Definitions

Definition 3 (Cf. [11], Solovyov) Given a commutative quantale Q, a quantale algebra is a quantale module (A, ∗) such that:

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

Q-algebras are equivalent to so called Q-quantales, where Q is a unital (commutative) quantale. Thus, Q-algebras can be seen as a fuzzification

• f quantales.

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• 1. Introduction: Quantale algebras

Definitions

Definition 3 (Cf. [11], Solovyov) Given a commutative quantale Q, a quantale algebra is a quantale module (A, ∗) such that:

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

Q-algebras are equivalent to so called Q-quantales, where Q is a unital (commutative) quantale. Thus, Q-algebras can be seen as a fuzzification

• f quantales.

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• 1. Introduction: Injective objects in categories

Definitions

Definition 4 Let C be a category and let M be a class of morphisms in

• C. We recall that an object S from C is M-injective in C provided that for

any morphism h: A − → B in M and any morphism f : A − → S in C there exists a morphism g : B − → S in C such that gh = f . A

✲

h ∈ M

✲ B

S

✛

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . g f

✲

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• 1. Introduction: M-injective hull

Definitions

Definition 5

M-essential morphism

A morphism η: A − → B in M is called M-essential if every morphism ψ: B − → C in C, for which the composite ψη is in M, is itself in M.

M-injective hull

An object H ∈ C is called an M-injective hull of an object S if H is M-injective and there exists an M-essential morphism η: S − → H. M-injective hulls are unique up to isomorphism. S✲ η ∈ M

✲ H

I

✛

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . g ∈ M

✲

f

✲

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• 1. Introduction: M-injective hull

Definitions

Definition 5

M-essential morphism

A morphism η: A − → B in M is called M-essential if every morphism ψ: B − → C in C, for which the composite ψη is in M, is itself in M.

M-injective hull

An object H ∈ C is called an M-injective hull of an object S if H is M-injective and there exists an M-essential morphism η: S − → H. M-injective hulls are unique up to isomorphism. S✲ η ∈ M

✲ H

I

✛

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . g ∈ M

✲

f

✲

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• 1. Introduction: Results on injective hulls of posets

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

Theorem 1 (Banaschewski B., Bruns G.) For a partially ordered set P, the following conditions are equivalent:

1

P is a complete lattice;

2

P is injective in posets with respect to the class of order embeddings;

3

P is a retract of every extension, i.e., for any order embedding j : P → R there is an order-preserving map k : R → P such that k ◦ j = idP;

4

P has no essential extensions, i.e., for any order embedding j : P → R the poset R is isomorphic to P.

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• 1. Introduction: Results on injective hulls of posets

Theorem 2 (Banaschewski B., Bruns G.) For a partially ordered set P, the following conditions are equivalent:

1

E is a MacNeille completion of P;

2

E is an injective hull of P;

3

E is an injective extension of P not containing any properly smaller such extension of P;

4

E is an essential extension of P not contained in any properly larger such extension of P.

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• 1. Introduction: Results on injective hulls of semilattices

1970, Bruns G., Lakser H., Injective hulls of meet-semilattices, Canadian Mathematical Bulletin

Theorem 3 (Bruns G., Lakser H.) A meet-semilattice S is injective iff it is a frame, i.e., it is complete and satisfies a ∧

• M =

a ∧ m | m ∈ M

• ,

(1) for all a ∈ S, M ⊆ S.

Distributive joins

We say that a subset M of a meet-semilattice S has a distributive join if (i) its supremum exists, and (ii) for all a ∈ S we have a ∧ M = a ∧ m | m ∈ M

• .

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• 1. Introduction: Results on injective hulls of semilattices

1970, Bruns G., Lakser H., Injective hulls of meet-semilattices, Canadian Mathematical Bulletin

Theorem 3 (Bruns G., Lakser H.) A meet-semilattice S is injective iff it is a frame, i.e., it is complete and satisfies a ∧

• M =

a ∧ m | m ∈ M

• ,

(1) for all a ∈ S, M ⊆ S.

Distributive joins

We say that a subset M of a meet-semilattice S has a distributive join if (i) its supremum exists, and (ii) for all a ∈ S we have a ∧ M = a ∧ m | m ∈ M

• .

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• 1. Introduction: Results on injective hulls of semilattices

1970, Bruns G., Lakser H., Injective hulls of semilattices. Bulletin

ID(S) = {A ⊆ S |A = A ↓; M ⊆ A has a distributive join ⇒

• M ∈ A}.

Theorem 4 (Bruns G., Lakser H.) The injective hull of a meet-semilattice S is (up to isomorphism) ID(S).

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• 1. Introduction: Results on injective hulls of semilattices

1970, Bruns G., Lakser H., Injective hulls of semilattices. Bulletin

ID(S) = {A ⊆ S |A = A ↓; M ⊆ A has a distributive join ⇒

• M ∈ A}.

Theorem 4 (Bruns G., Lakser H.) The injective hull of a meet-semilattice S is (up to isomorphism) ID(S).

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• 1. Introduction: Results on injective hulls of certain

S-systems over a semilattice

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

Theorem 5 (Johnson C.S., J.R., McMorris F.R.) The injective hull of an S-system MS is (up to isomorphism) 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 trivial injectives.

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• 1. Introduction: Results on injective hulls of 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 submultiplicative

• rder-preserving mappings, i.e., an order-preserving mapping φ : A → B of

po-monoids satisfying φ(1) = 1, φ(a) · φ(b) ≤ φ(a · b), for all a, b ∈ A.

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• 1. Introduction: Results on injective hulls of 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 (S, ·) is injective iff it is a quantale, i.e., it is complete and satisfies a ·

• M =

a · m | m ∈ M

• ,

(2) for all a ∈ S, M ⊆ S..

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• 2. Injective constructions for posemigroups (Definitions)

Category Pos≤

A posemigroup (S, ·, ≤) is both a semigroup (S, ·, ) and a poset (S, ≤) such that for any a, b, c, d ∈ S, a b, c ≤ d = ⇒ a · c ≤ b · d.

Morphisms in posemigroups: order-preserving submultiplicative mappings (subhomomorphisms), i.e., f(a1) · f(a2) ≤ f(a1 · a2) for all a1, a2 ∈ S.

We denote the category of posemigroups with subhomomorphisms as morphisms by Pos≤.

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• 2. Injective constructions for posemigroups (Definitions)

Category Pos≤

A posemigroup (S, ·, ≤) is both a semigroup (S, ·, ) and a poset (S, ≤) such that for any a, b, c, d ∈ S, a b, c ≤ d = ⇒ a · c ≤ b · d.

Morphisms in posemigroups: order-preserving submultiplicative mappings (subhomomorphisms), i.e., f(a1) · f(a2) ≤ f(a1 · a2) for all a1, a2 ∈ S.

We denote the category of posemigroups with subhomomorphisms as morphisms by Pos≤.

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• 2. Injective constructions for posemigroups (Definitions)

Category Pos≤

A posemigroup (S, ·, ≤) is both a semigroup (S, ·, ) and a poset (S, ≤) such that for any a, b, c, d ∈ S, a b, c ≤ d = ⇒ a · c ≤ b · d.

Morphisms in posemigroups: order-preserving submultiplicative mappings (subhomomorphisms), i.e., f(a1) · f(a2) ≤ f(a1 · a2) for all a1, a2 ∈ S.

We denote the category of posemigroups with subhomomorphisms as morphisms by Pos≤.

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• 2. Injective constructions for posemigroups (Definitions)

Embeddings in Pos≤

Let ε≤ be the class of morphisms e : S − → T in the category Pos≤ which satisfy the following condition: e(a1) · . . . · e(an) ≤ e(a) = ⇒ a1 · . . . · an ≤ a, for all a1, a2, . . . , an ∈ S.

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• 2. Injective constructions for posemigroups (Examples)

An example (Rosenthal)

Let (S, ·, ≤) be a posemigroup, P(S) the set of all of the downsets in S. Then (P(S), ⊙, ⊆) is a posemigroup, where X ⊙ Y = (X · Y ) ↓= {a ∈ S | a ≤ x · y, for some x ∈ X, y ∈ Y }, for all X, Y ⊆ S. Moreover, P(A) is a complete lattice whose joins are unions. Consequently, we obtain that (P(S), ⊙, ⊆) is a quantale.

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• 2. Injective constructions for posemigroups (Injectives)

Theorem 8 (2014 Zhang, Laan) For a posemigroup S, the following statements are equivalent:

1 S is ε≤-injective in Pos≤, 2 S is a quantale. Jan Paseka (MU)

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• 2. Injective constructions for posemigroups (Constructions)

Definition: Nucleus

A closure operator j on a posemigroup S is called a nucleus if it is a

• subhomomorphism. We denote Sj = {a ∈ S | a = j(a)}.

Proposition

Let (S, ·, ≤) be a posemigroup, j a nucleus on it. Then (Sj, ·j, ≤) is again a posemigroup equipped with the multiplication and induced order as a ·j b = j(a · b), for any a, b ∈ S. In addition, if S is a quantale, then (Sj, ·j, ≤) is a quantale as well.

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• 2. Injective hulls for posemigroups (Constructions)

the nucleus cl: Zhang, Laan 2014, Xia, Zhao, Han 2017

Let (S, ·, ≤) be a posemigroup, D ∈ P(S). Define cl(D) := {x ∈ S | (∀a, c ∈ S1, b ∈ S) a · D · c ⊆ b ↓= ⇒ a · x · c ≤ b}, where S1 is the monoid obtained from the semigroup S by externally adjoining the identity element 1. Then cl : P(S) → P(S) is a nucleus on P(S) satisfying cl(x ↓) = x ↓, ∀x ∈ S.

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• 2. Injective hulls for posemigroups

Theorem

Theorem 9 (Zhang, Laan 2014, Xia, Zhao, Han 2017)

Let (S, ·, ≤) be a posemigroup, cl: P(S) − → P(S) be defined as above. Then (P(S)cl, ⊙cl, ⊆) is the ε≤-injective hull of the posemigroup S.

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• 3. Injective constructions for S-semigroups (Definitions)

(S, ⊗) is always a posemigroup

A posemigroup (A, ·, ≤) together with a mapping S × A → A (under which a pair (s, a) maps to an element of A denoted by s ∗ a) is called an S-semigroup, denoted by SA, or simple A, if for any a, b ∈ A, s, t ∈ S,

1

s ∗ (a · b) = (s ∗ a) · b = a · (s ∗ b),

2

(s ⊗ t) ∗ a = s ∗ (t ∗ a), fulfilling a b, s ≤ t = ⇒ s ∗ a ≤ t ∗ b.

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• 3. Injective constructions for S-semigroups (Definitions)

Morphisms in Ssgr≤

An order-preserving mapping f : SA − → SB of S-semigroups is called a subhomomorphism if it is both submultiplicative in posemigroups, i.e., f(a1) · f(a2) ≤ f(a1 · a2) for all a1, a2 ∈ A, and S-submultiplicative in S-posets, i.e., s ∗ f(a) ≤ f(s ∗ a) for all a ∈ A, s ∈ S.

We denote the category of S-semigroups with subhomomorphisms as morphisms by Ssgr≤.

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• 3. Injective constructions for S-semigroups (Definitions)

Embeddings in Ssgr≤

Let ε0 be the class of morphisms e :

SA −

→

SB in the category Ssgr≤ which

satisfy the following conditions: s ∗ (e(a1) · . . . · e(an)) ≤ e(a) = ⇒ s ∗ (a1 · . . . · an) ≤ a, and e(a1) · . . . · e(an) ≤ e(a) = ⇒ a1 · . . . · an ≤ a, for all a1, a2, . . . , an, a ∈ A, s ∈ S. Then ε ⊆ ε0, where ε is the class of all S-semigroup homomorphisms that are

• rder-embeddings.

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• 3. Injective constructions for S-semigroups (Definitions)

A new quantale-like structure

An S-semigroup quantale is an S-semigroup (SA, ·, ∗) such that

1

(A, ·, ) is a quantale;

2

s ∗ M = {s ∗ m | m ∈ M}, for every s ∈ S, M ⊆ A.

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• 3. Injective constructions for S-semigroups

An example

Let (SA, ·, ∗) be an S-semigroup, P(A) the set of all of the downsets in A. Then (SP(A), ⊙, ⊛, ⊆) is an S-semigroup, where X ⊙ Y = (X · Y ) ↓= {a ∈ A | a ≤ x · y, for some x ∈ X, y ∈ Y }, s ⊛ X = (s ∗ X) ↓= {a ∈ A | a ≤ s ∗ x, for some x ∈ X}, for all X, Y ⊆ A, s ∈ S. Moreover, SP(A) is a complete lattice whose joins are

• unions. Consequently, we obtain that (SP(A), ⊙, ⊛, ⊆) is an S-semigroup

quantale.

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• 3. Injective constructions for S-semigroups (Injectives)

Theorem 8 (2017 Zhang, Paseka) For an S-semigroup SA, the following statements are equivalent:

1

SA is ε0-injective in Ssgr≤,

2

SA is ε-injective in Ssgr≤,

3

SA is an S-semigroup quantale.

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• 3. Injective constructions for S-semigroups (Constructions)

Definition: Nucleus

A closure operator j on an S-semigroup SA is called a nucleus if it is a

• subhomomorphism. We denote Aj = {a ∈ A | a = j(a)}.

Proposition

Let (SA, ·, ∗) be an S-semigroup, j a nucleus on it. Then (Aj, ·j, ∗j) is again an S-semigroup equipped with the multiplication and action as a ·j b = j(a · b), s ∗j a = j(s ∗ a), for any a, b ∈ A, s ∈ S. In addition, if A is an S-semigroup quantale, then (Aj, ·j, ∗j) is an S-semigroup quantale as well.

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• 3. Injective hulls for S-semigroups (Constructions)

the nucleus cl:

Let (SA, ·, ∗) be an S-semigroup, D ∈ SP(A). Define cl(D) := {x ∈ A | (∀a, c ∈ A1, b ∈ A, s ∈ S) a · D · c ⊆ b ↓= ⇒ a · x · c ≤ b, s ∗ D ⊆ b ↓= ⇒ s ∗ x ≤ b}, where A1 is the monoid obtained from the semigroup A by externally adjoining the identity element 1. Then cl : P(A) → P(A) is a nucleus on

SP(A) satisfying cl(x ↓) = x ↓, ∀x ∈ A.

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• 3. Injective hulls for S-semigroups

Theorem

Theorem 9 (2017 Zhang, Paseka)

Let (SA, ·, ∗) be an S-semigroup, cl: SP(A) − →

SP(A) be defined as

• above. Then (P(A)cl, ⊙cl, ⊛cl, ⊆) is the ε0-injective hull of the

S-semigroup SA.

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Conclusion Remarks

We have described injectives in the category Ssgr≤ of S-semigroups and showed that every S-semigroup has an ε0-injective hull. Based on these results the next step in the future would be to obtain corresponding results in the category of residuated S-semigroups.

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References

[1] Ad´ amek J., Herrlich H. and Strecker G. E., Abstract and concrete categories: The joy

• f 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., 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 a semilattice, 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

• rdered 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, 2012, 85, 571-576. [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, Theory

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[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.

Jan Paseka (MU)

• 10. 8. 2018

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Thank you!

Jan Paseka (MU)

• 10. 8. 2018

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