On the Category of Pointed G-Sets

Home , Pointed set

Int. Journal of Math. Analysis, Vol. 7, 2013, no. 2, 65 - 71

G. Muhiuddin

Department of Mathematics University of Tabuk P. O. Box 741, Tabuk 71491, Saudi Arabia [email protected]

Abdullah M. Al-roqi

Department of Mathematics University of Tabuk P. O. Box 741, Tabuk 71491, Saudi Arabia

M. Irfan

Department of Mathematics Aligarh Muslim University Aligarh - 202 002, India

Abstract In this paper, we obtain that the category of pointed G-sets has kernels and cokernels. Further, we prove that the category of pointed G-sets is normal and factorizable.

Mathematics Subject Classiﬁcation: 18A05, 20C05

Keywords: pointed G-sets, pointed G-morphisms, category of pointed G- sets

1. Introduction

Pointed sets may be regarded as a rather simple algebraic structure. In the sense of universal algebra, these are structures with a single nullary operation which picks out the base point. Pointed sets are mainly used as illustrative examples in the study of universal algebra as algebra with a single constant operator. This operator takes every element in the algebra to a unique constant, which is known as a base point. Any homomorphism between two algebras 66 G. Muhiuddin, Abdullah M. Al-roqi and M. Irfan preserves base points (taking the base point of the domain algebra to the base point of the codomain algebra). Motivated by the idea of pointed sets and pointed mappings, pointed G-sets and pointed G-morphisms have been stud- ied in [2,3].

Henceforth the category G-Sets*, of pointed G-sets has been constructed by taking into account pointed G-sets as the objects of the category and pointed G-morphisms as the morphisms of the category. The above mentioned paper deals with the behaviour of some special morphisms such as monomor- phisms, epimorphisms, sections and retractions in the category G-Sets*. As a follow up, in the present analysis we obtain some interesting results in this category by proving that the category G-Sets* possesses kernels and cokernels. The existence of these notions is used in achieving that the category G-Sets* is normal and factorizable.

2. Preliminaries

We begin with the following deﬁnitions and results that will be needed in the sequel [2,3,6]:

Definition 2.1. Let G be a group and X be a set. Then X is said to be a G-set if there exists a mapping φ : G × X → X such that for all a, b ∈ G and x ∈ X the following conditions are satisfied: (i) φ(ab, x)=φ(a, φ(b, x)) (ii) φ(e, x)= x, where e is the identity of G. The G-set X defined above will be denoted by the pair (X, φ).

For the sake of convenience, one can denote φ(a, x)asax. Under this notation, above conditions become (i) (ab)x = a(bx) (ii) ex = x.

Deﬁnition 2.2. Let X and Y be two G-sets. Then a mapping f : X → Y is called a G-morphism from X to Y if f(ax)=af(x) for all a ∈ G, x ∈ X.

Deﬁnition 2.3. Let (X, φ) be a G-set. Then a subset A of X is called a G-subset of X if (A, φ) is also a G-set.

Deﬁnition 2.4. A pointed set (X, x) is said to be a pointed G-set if there exists a mapping φ : G × X → X such that

(i) (X, φ) is a G-set,

(ii) φ(g, x)=x for all g ∈ G. On the category of pointed G-sets 67

Deﬁnition 2.5. Let (X, x) be a pointed G-set. Then a pointed G-subset of (X, x) is an ordered pair (A, x), where A is a G-subset of X.

Deﬁnition 2.6. Let (X, x) and (Y,y) be two pointed G-sets. Then a mapping f :(X, x) → (Y,y) is called a pointed G-morphism if (i) f is a G-morphism i.e., f(ax)=af(x) for all a ∈ G, x ∈ X, (ii) f(x)=y. Deﬁnition 2.7. Let α :(X, x) → (Y,y) be a morphism in G-Sets*. Then the subset Ker(α)={x | α(x)=y} of X is called the kernel of the morphism α.

Lemma 2.1[6]. Let (X, φ) be a G-set. Then

(i) for any x, y ∈ X, a relation ∼G on X deﬁned by x∼Gy ⇔ y = φ(g, x) for some g ∈ G, is an equivalence relation,

(ii) the set X/∼G of all G-equivalence classes is a G-set. Deﬁnition 2.8. Let α :(X, x ) → (Y,y ) be a morphism in G-Sets* and ∼G be a G-equivalence relation on Y . Then the pointed G-set (Y/∼G, [y ]) is called the co-kernel of α and is denoted by Coker(α).

3. Main Results

Proposition 3.1. Let α :(X, x) → (Y,y) be a pointed G-morphism. Then Ker(α) is a pointed G-subset of X in G-Sets*.

Proof. In order to show that Ker(α) is a pointed G-subset of X, deﬁne a mapping φ : G×Ker(α) → Ker(α)byφ(a, x)=ax for all a ∈ G, x ∈ Ker(α).

Let x ∈ Ker(α). Then we have α(x)=y implying thereby aα(x)=ay for all a ∈ G. Therefore, α(ax)=y yielding ax ∈ Ker(α) and so the mapping φ is well deﬁned.

For any a, b ∈ G and x ∈ X, we get φ(ab, x)=(ab)x = a(φ(b, x)) = φ(a, φ(b, x)). Also for the identity element e ∈ G, one gets φ(e, x)=x. There- fore Ker(α) is a G-subset of X.

Further, since f is a pointed G-morphism, then f(x)=y implies x ∈ Ker(α) and also for any a ∈ G, we get φ(a, x)=ax = x which amounts to say that (Ker(α),x) is a pointed G-subset of (X, x). 68 G. Muhiuddin, Abdullah M. Al-roqi and M. Irfan

Theorem 3.1. The category G-Sets* has kernels.

Proof. Let f :(X, x) → (Y,y) be a morphism in G-Sets*. Consider the set K = {x | f(x)=y}⊆X, then in view of Proposition 3.1, K is a pointed G-subset. Let i :(K, x) → (X, x) be an inclusion morphism in G-Sets*. We claim that i :(K, x) → (X, x) is the kernel of f :(X, x) → (Y,y)inG-Sets*.

For any x ∈ K, one gets (f ◦ i)(x)=f(i(x)) = f(x)=y = OKY (x) which implies f ◦ i = OKY .

Now, for any object (Z, z) ∈G-Sets*, let α :(Z, z) → (X, x)bea morphism in G-Sets* such that f ◦ α = OZY .

For any z ∈ Z, one gets (f ◦ α)(z)=OZY (z) implying thereby f(α(z)) = y. Henceforth α(z) ∈ K which yields Im(α) ⊆ K. Therefore we can deﬁne a mapping η :(Z, z) → (K, x)byη(z)=α(z) for all z ∈ Z.

For any z ∈ Z and a ∈ G, we get η(az)=α(az)=aα(z)=a(η(z)) showing that η is a G-morphism and also η(z)=α(z)=x which amounts to say that η :(Z, z) → (K, x) is a morphism in G-Sets*.

Moreover, for any z ∈ Z, we have (i ◦ η)(z)=i(η(z)) = i(α(z)) = α(z) which implies i ◦ η = α.

Finally, in order to show that η is unique, suppose there is another morphism ξ :(Z, z) → (K, x)inG-Sets* such that i ◦ ξ = α. Then for any z ∈ Z, we have (i ◦ ξ)(z)=α(z) implying i(ξ(z)) = η(z) which gives ξ(z)=η(z). Therefore ξ = η and hence the result follows.

Theorem 3.2. The category G-Sets* has co-kernels.

Proof. Let f :(X, x) → (Y,y) be a morphism in G-Sets*. Let R be a relation on Y such that for any y1,y2 ∈ Y , y1Ry2 ⇔ y1,y2 ∈ Im(f). Consider a smallest equivalence relation R¯ on Y containing R such that for any y1,y2 ∈ Y , y1Ry¯ 2 ⇔ y1 = y2 or y1,y2 ∈ Im(f). The equivalence classes [y] under R¯ are given by ⎧ ⎨ [f(x)] = Imf if y = f(x) ∈ Imf [y]=⎩ {y} if y ∈ Imf.

Clearly, Y/R¯ is a pointed G-set with the base point [y]. On the category of pointed G-sets 69

Since y = f(x), it follows that y ∈ Im(f) yielding thereby [y]=Im(f). Deﬁne a natural mapping p :(Y,y) → (Y/R,¯ [y]) by p(y)=[y] for all y ∈ Y in G-Sets*. We claim that p :(Y,y) → (Y/R,¯ [y]) is the co-kernel of f :(X, x) → (Y,y)inG-Sets*.

For any x ∈ X, one gets (p ◦ f)(x)=p(f(x)) = [f(x)] = Im(f)=[y]= OXY/R¯(x) which implies p ◦ f = OXY/R¯.

For any object (Z, z) ∈G-Sets*, let q :(Y,y) → (Z, z) be a morphism in G-Sets* such that q ◦ f = OXZ.

For any x ∈ X, one gets (q ◦ f)(x)=OXZ(x) implying thereby q(f(x)) = z.Thusf(x) ∈ Ker(q) which gives Im(f) ⊆ Ker(q). Therefore we can deﬁne a mapping η : Y/R¯ → Z by η([y]) = q(y) for all y ∈ Y.

Let [y1]=[y2] which gives y1 ∼R¯ y2. Therefore y1 = y2 or y1,y2 ∈ Im(f).

Now, if y1 = y2, then q(y1)=q(y2) which implies η([y1]) = η([y2]). Again, if y1,y2 ∈ Im(f) ⊆ Ker(q), then q(y1)=z = q(y2) which implies η([y1]) = η([y2]). Consequently η is well deﬁned.

For any [y] ∈ Y/R¯ and a ∈ G, one gets η(a[y]) = η([ay]) = q(ay)= a(q(y)) = a(η([y])) which shows that η is a G-morphism and also η([y]) = q(y)=z which in turn yields that η :(Y/R,¯ [y]) → (Z, z) is a morphism in G-Sets*.

Moreover, for any y ∈ Y , we have (η ◦ p)(y)=η(p(y)) = η([y]) = q(y) which implies η ◦ p = q.

Finally, in order to prove that η is unique, suppose there is another morphism ξ :(Y/R,¯ [y]) → (Z, z)inG-Sets* such that ξ ◦ p = q. Then for any y ∈ Y , we have (ξ ◦ p)(y)=q(y) implying thereby ξ(p(y)) = η([y]). Therefore ξ([y]) = η([y]) which implies ξ = η and hence the result follows.

Theorem 3.3. In the category G-Sets* every monomorphism is a kernel.

Proof. Let f :(X, x) → (Y,y) be a monomorphism in G-Sets*, then f is injective in G-Sets* [2, Corollary 3.1]. Thus X Im(f) ⊆ Y . In view of Theorem 3.2, p :(Y,y) → (Y/R,¯ [y]) is the co-kernel of f :(X, x) → (Y,y)in the category G-Sets* and we have p ◦ f = OXY/R¯. We claim that f :(X, x ) → (Y,y) is the kernel of p :(Y,y) → (Y/R,¯ [y]). 70 G. Muhiuddin, Abdullah M. Al-roqi and M. Irfan

Let α :(Z, z ) → (Y,y ) be a morphism in G-Sets* such that p◦α = OZY/R¯. Then for any z ∈ Z, we have (p ◦ α)(z)=OZY/R¯(z) yielding thereby p(α(z)) = [y]=Im(f). Thus [α(z)] = Im(f) which gives α(z) ∈ Im(f) implying thereby Im(α) ⊆ Im(f). Therefore we can deﬁne a mapping η :(Z, z) → (X, x)byη(z)=x where f(x)=α(z).

Here, it may be noted that x is unique in view of the injectivity of f, therefore the mapping η is well deﬁned.

For any a ∈ G, we have α(az)=a(α(z)) = a(f(x)) = f(ax) which amounts to say that η(az)=ax = a(η(z)) implying thereby that η is a G-morphism and also α(z)=y = f(x) yields that η(z)=x. Consequently, η is a morphism in G-Sets*. Moreover, for any z ∈ Z, we get (f ◦ η)(z)=f(η(z)) = f(x)=α(z) implying thereby f ◦ η = α.

Finally, in order to prove that η is unique, suppose there is another morphism ξ :(Z, z) → (X, x)inG-Sets* such that f ◦ξ = α. Then f ◦ξ = f ◦η which implies ξ = η as f is left cacellable. This completes the proof.

Theorem 3.4. The category G-Sets* is normal.

Proof. Since, the category G-Sets* has zero object, therefore the proof follows from Theorem 3.1, Theorem 3.2 and Theorem 3.3.

Theorem 3.5. The category G-Sets* is factorizable.

Proof. In view of the above Theorem 3.4, the category G-Sets* is normal, therefore it is factorizable [1, Theorem 4.8].

Acknowledgments. The authors would like to express their sincere thanks to the anonymous referees for their useful suggestions. This research is supported by the Deanship of Scientiﬁc Research, University of Tabuk, Tabuk, Saudi Arabia.

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Received: August, 2012