Theory and Applications of Categories, Vol. 32, No. 7, 2017, pp. 254–257.

A NOTE ON INJECTIVE HULLS OF POSEMIGROUPS

CHANGCHUN XIA, SHENGWEI HAN, BIN ZHAO

Abstract. In this note, we prove the existence of E_{≤} -injective hulls in the category PoSgr_{≤} of posemigroups and their submultiplicative order-preserving maps; here E_{≤} denotes the class of those morphisms h: A → B for which h(a_{1})··· h(a_{n}) ≤ h(a) always implies a_{1}··· a_{n} ≤ a. The result of this note subsumes the results given by Lambek et al. (2012) and by Zhang and Laan (2014).

1. Introduction

The injective hull, though not under that name, was first obtained in [Baer (1940)]. The bestknown early discussion of injective hulls is in [Eckmann and Schopf (1953)], who introduced them as maximal essential extensions. Banaschewski and Bruns [1967] characterized the injective hulls in the category of posets. In 1970, Bruns and Lakser characterized the injective hulls in the category of semilattices. As we all know, the injective objects in the category of pomonoids with pomonoid homomorphisms are trivial [Lambek et al. (2012)]. Hence, Lambek et al. changed a few of the definitions of pomonoid homomorphisms, chose submultiplicative order-preserving maps as morphisms of the category PoMon_{≤} of pomonoids to investigate the injectivity in PoMon_{≤}, where a submultiplicative map between pomonoids is a map f: (A,·,≤) → (B,∗,≤) such that f(a) ∗ f(a′) ≤ f(a · a′) for all a,a′∈ A. Also, they proved that every pomonoid has an injective hull and gave the concrete form of the injective hulls. Based on the work of Lambek et al., dropping out the unit, Zhang and Laan intended to consider the injective hulls in the category PoSgr_{≤} of posemigroups, and constructed the injective hulls for a certain class of posemigroups with respect to a class of order embedding. However, for the question whether every posemigroup has an E_{≤} -injective hull in PoSgr_{≤}, Zhang and Laan did not give an answer. In this note, we shall give an affirmative answer and obtain the concrete form of injective hulls for general posemigroups. Particularly, when a posemigroup S is a pomonoid, the injective hull of S in PoSgr_{≤} we obtain is exactly the injective hull of S in PoMon_{≤} Lambek et al. gave.

A partially ordered semigroup (posemigroup for short) is a semigroup (S,·) with a partial order ≤ on S which is compatible with the partial order ≤, that is, x ≤ y ⇒ x·z ≤ y·z and z·x ≤ z · y for any z in S. A quantale is a complete lattice Q with an associative binary operation & satisfying a&(⋁_{i∈I} b_{i}) = ⋁_{i∈I}(a&b_{i}) and (⋁_{i∈I} b_{i})&a = ⋁_{i∈I}(b_{i}&a) for any a ∈ Q and {b_{i}}_{i∈I} ⊆ Q. A quantic nucleus j on a quantale Q is a submultiplicative closure operator. It is easy to show that j(a&b) = j(a&j(b)) = j(j(a)&b) = j(j(a)&j(b)) for all a,b ∈ Q. Furthermore, we can also prove that the set Q_{j} of fixed points of j is a quantale with a ⊗ b = j(a&b) and

We would like to thank the National Natural Science Foundation of China (11531009), the Natural Science Program for Basic Research of Shaanxi Province, China ( 2015JM1020) and the Fundamental Research Funds for the Central Universities (GK201501001).

Received by the editors 2017-01-19 and, in final form, 2017-02-01.

Transmitted by Walter Tholen. Published on 2017-02-08.

2010 Mathematics Subject Classification: 06F05, 06F07, 08B30.

Key words and phrases: partially ordered semigroup, quantale, quantic nucleus, injective hull.

\begin{matrix}c \\ ◯ \end{matrix} Changchun Xia, Shengwei Han, Bin Zhao , 2017. Permission to copy for private use granted.

⋁_{i∈I}^{Q_{j}} a_{i} = j(⋁_{i∈I}^{Q} a_{i}). Clearly, every quantale is necessarily a posemigroup. For a given semigroup

(S,·), we denote by P(S) the set of all subsets of S. Clearly, (P(S),•) forms a quantale under the inclusive order, where A•B = {a·b: a ∈ A,b ∈ B}. When A = {a}, we write A•B and B•A simply as a · B and B · a.

Let C be a category and let M be a class of morphisms in C. Then an object S in 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 such that g∘h = f. A morphism η: A → B in M is called M -essential provided that every morphism ψ: B → C in C for which the composite ψ ∘η is in M is itself in M. An object H ∈ C is said to be an M -injective hull of an object of S provided that H is M -injective and there exists an M -essential morphism S → H.

For notions and concepts concerned, but not explained, please refer to [Lambek et al. (2012)], [Zhang and Laan (2014)] and [Adámek et al. (2004)].

2. Main results

In this section, we shall show that every posemigroup has an E_{≤} -injective hull, and give its concrete form. Firstly, we review some main results on injective hulls in the category PoMon_{≤} and PoSgr_{≤}.

2.1. Theorem. [Lambek et al. (2012)] A pomonoid is injective (with respect to embeddings) if and only if it is a quantale.

2.2. Theorem. [Lambek et al. (2012)] The map η(x ↦ x ↓): A → Q(A) is an essential embedding and thus is an embedding into an injective hull.

2.3. Theorem. [Zhang and Laan (2014)] Let (S,·,≤) be a posemigroup. Then S is E_{≤} -injective in PoSgr_{≤} if and only if S is a quantale.

2.4. Theorem. [Zhang and Laan (2014)] Let S be a posemigroup such that cl(s ↓) = s ↓ for every s ∈ S. Then D(S) is an E_{≤} -injective hull in PoSgr_{≤}.

In what follows we shall prove our main result.

Let (S,·,≤) be a posemigroup and a ∈ S. Then we denote by a↓ the set {b ∈ S: b ≤ a}. For any subset X of S, we define its closure by X^{⋆}

X^{⋆} ≜ X^{ul}∩ X^{L}∩ X^{R}∩ X^{T}

where

X^{ul} ≜ {s ∈ S: ∀ b ∈ S,X ⊆ b↓⇒ s ≤ b}

X^{L} ≜ {s ∈ S: ∀ a,b ∈ S,X · a ⊆ b↓⇒ s · a ≤ b}
X^{R}≜ {s ∈ S: ∀ a,b ∈ S,a · X ⊆ b↓⇒ a · s ≤ b}
X^{T} ≜ {s ∈ S: ∀ a,b,c ∈ S,a · X · c ⊆ b↓⇒ a · s · c ≤ b}

It is easy to verify that X^{⋆} = X^{T} when S is a pomonoid.

2.5. Lemma. [Xia et al. (2016)] Let (S,·,≤) be a posemigroup. Then (·)^{⋆} is a quantic nucleus on the power-set quantale (P(S),•), that is, (S^{⋆},⊗) is a quantic quotient of (P(S),•), where S^{⋆} = {A ∈ P(S) : A^{⋆} = A}, X ⊗ Y = (X • Y)^{⋆}.

In [Xia et al. (2016)], we prove that (·)^{⋆} is also topological, that is, {x}^{⋆} = x↓ for all x ∈ S. We define now a map η_{S}: S → S^{⋆} as follows η_{S}(x) = {x}^{⋆} for all x ∈ S. In [Xia et al. (2016)], we prove that (η_{S},S^{⋆}) is the least quantale completion of S. In the following we shall see that η_{S} is in fact E_{≤} -essential in PoSgr_{≤}, that is, S^{⋆}is an E_{≤} -injective hull of S in PoSgr_{≤}.

2.6. Theorem. Let (S,·,≤) be a posemigroup. Then S^{⋆}is an E_{≤} -injective hull of S in PoSgr_{≤}.

Proof. One can easily see that (x↓)^{⋆} = x↓ for any x ∈ S. In the following we shall prove that η_{S} is an E_{≤} -essential morphism in PoSgr_{≤}. Clearly, η_{S} belongs to E_{≤}. By Theorem 2.3, S^{⋆} is E_{≤} - injective. It remains to show that η_{S} is E_{≤} -essential. Let ψ: S^{⋆} → (B,≤,∗) be a morphism in PoSgr_{≤} such that ψ ∘ η_{S} ∈ E_{≤}. We shall verify that ψ ∈ E_{≤}. Suppose that ψ(X_{1}) ∗ ψ(X_{2}) ∗ ··· ∗ ψ(X_{n}) ≤ ψ(Y) in B, where X_{1},X_{2},···,X_{n},Y ∈ S^{⋆}. We need to show that X_{1}⊗X_{2}⊗··· ⊗X_{n} ⊆ Y. In fact, we only need to prove that X_{1}• X_{2}• ··· • X_{n} ⊆ Y^{ul}∩ Y^{L}∩ Y^{R}∩ Y^{T}.

(1) X_{1}• X_{2}• ··· • X_{n} ⊆ Y^{ul}.

Assume that x_{1} ∈ X_{1},···,x_{n} ∈ X_{n}. We let b ∈ S and Y ⊆ b↓. Then we have

(ψ ∘ η_{S})(x_{1}) ∗ ··· ∗ (ψ ∘ η_{S})(x_{n}) = ψ(x_{1}↓) ∗ ··· ∗ ψ(x_{n}↓)
≤ ψ(X_{1}) ∗ ··· ∗ ψ(X_{n})
≤ ψ(Y)
≤ ψ(b↓)
= (ψ ∘ η_{S})(b).

It follows from the case that ψ ∘η_{S} ∈ E_{≤} that x_{1}·x_{2}· ····x_{n} ≤ b. Hence, X_{1}•X_{2}•··· •X_{n} ⊆ b↓, which implies X_{1}• X_{2}• ··· • X_{n} ⊆ Y^{ul}.

(2) X_{1}• X_{2}• ··· • X_{n} ⊆ Y^{L}.

Suppose that x_{1} ∈ X_{1},···,x_{n} ∈ X_{n}. We let a,b ∈ S and Y · a ⊆ b↓. Then we have

(ψ ∘ η_{S})(x_{1}) ∗ ··· ∗ (ψ ∘ η_{S})(x_{n}) = ψ(x_{1}↓) ∗ ··· ∗ ψ(x_{n}↓) ∗ ψ(a↓)
≤ ψ(X_{1}) ∗ ··· ∗ ψ(X_{n}) ∗ ψ(a↓)
≤ ψ(Y) ∗ ψ(a↓)
≤ ψ(Y ⊗ a↓)
≤ ψ((Y · a)^{⋆})
≤ ψ(b↓)
= (ψ ∘ η_{S})(b),

It follows from the case that ψ ∘ η_{S} ∈ E_{≤} that x_{1}· x_{2}· ···· x_{n}· a ≤ b. Hence, (X_{1}• X_{2}• ··· • X_{n}) · a ⊆ b↓, which implies X_{1}• X_{2}• ··· • X_{n} ⊆ Y^{L}.

(3) Similar to (2), we can prove that X_{1}• X_{2}• ··· • X_{n} ⊆ Y^{R}and X_{1}• X_{2}• ··· • X_{n} ⊆ Y^{T}.

By (1)-(3), we conclude that X_{1}•X_{2}•··· •X_{n}⊆ Y^{ul}∩Y^{L}∩Y^{R}∩Y^{T}, that is, X_{1}•X_{2}•··· •X_{n}⊆ Y. Therefore, S^{⋆} is an E_{≤} -injective hull of S in PoSgr_{≤}.

2.7. Remark. S^{⋆} is the injective hull of S in PoMon_{≤} whenever S is a pomonoid.

References

J. Adámek, H. Herrlich and G.E. Strecker (2004), Abstract and Concrete Categories: The joy of cats, Wiley, New York.

B. Banaschewski and G. Bruns (1967). Categorical characterization of the MacNeille completion. Archiv der Mathematik. 18, 369–377.

R. Baer (1940). Abelian groups which are direct summands of every containing group. Bull. Mmer. Math. Soc. 46, 800–806.

G. Bruns, H. Lakser (1970), Injective hulls of semilattices. Canad. Math. Bull. 13, 115–118.

B. Eckmann and A. Schopf (1953), Über injektive Moduln. Archiv der Mathematik 4, 75–78.

J. Lambek, M. Barr, J.F. Kennison and R. Raphael (2012), Injective hulls of partially ordered monoids. Theory Appl. Categ. 26, 338–348.

C.C. Xia, B. Zhao and S.W. Han (2016), Some futher results of quantale completions of ordered semigroup. Submitted to Acta Mathematica Sinica, Chinese Series.

X. Zhang, V. Laan (2014), Injective hulls for posemigroups. Proc. Est. Acad. Sci. 63, 372–378.

Department of Mathematics, Shaanxi Normal University, Xi’an 710119, China

Email: xiachangchun@snnu.edu.cn
      hansw@snnu.edu.cn
      zhaobin@snnu.edu.cn
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