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On the Category of Pointed G-Sets Int. Journal of Math. Analysis, Vol. 7, 2013, no. 2, 65 - 71 On the Category of Pointed G-Sets 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 op- erator. 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 mor- phism 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]).
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