EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 6, No. 4, 2013, 469-484 ISSN 1307-5543 – www.ejpam.com Vector Space–groupoids Mihai Ivan Department of Educational Sciences, West University of Timi̧ soara, Timi̧ soara, Romania Abstract. We define the notion of vector space-groupoid. The main purpose of this paper is to give the basic properties of vector space-groupoids. 2010 Mathematics Subject Classifications: 20L13, 20L99 Key Words and Phrases: groupoid, group-groupoid, vector space-groupoid 1. Introduction In the category theoretical approach, a groupoid is a small category in which every mor- phism is an isomorphism [6]. The concept of groupoid was first introduced by H. Brandt [1] and it is developed by P. J. Higgins in [6]. The topological and differentiable versions of the groupoids were defined by C. Ehresmann [5]. The notion of group-groupoid was defined by R. Brown and Spencer in the paper [4]. In this paper, the group-groupoid is extended to notion of vector space-groupoid. Another algebraic concept considered in this paper is the vector groupoid. This new mathematical structure was defined by V. Popuţa and Gh. Ivan [13, 14]. The groupoids, group- groupoids and their generalizations (topological groupoids, Lie groupoids etc.) are mathematical structures that have proved to be useful in many areas of science (see for instance [2, 7, 9–12, 15]. The paper is organized as follows. In Section 2 we present some concepts and main results related to groupoids and group-groupoids [4]. In Section 3 we introduce the concept of vector space-groupoid. This is viewed as a groupoid object in the category of vector spaces. The useful properties of vector space-groupoids are established. Finally, we prove that each vector space-groupoid is a vector groupoid. Email address: ivan@math.uvt.ro http://www.ejpam.com 469 c© 2013 EJPAM All rights reserved. M. Ivan / Eur. J. Pure Appl. Math, 6 (2013), 469-484 470 2. Preliminaries about Group-groupoids We begin with the presentation of some necessary backgrounds on groupoids (for further details see e.g. [8, 9]). Definition 1 ([9]). A groupoid G over G0 is a pair (G, G0) of sets endowed with two surjective maps α,β : G→ G0 (source and target), a partially binary operation (multiplication) m : G(2) := {(x , y) ∈ G × G|β(x) = α(y)} → G, (x , y) → m(x , y) := x · y, (G(2) is the set of composable pairs), an injective map ε : G0 → G (inclusion map) and a map i : G → G, x → i(x) := x−1 (inversion). These maps must verify the following conditions: (G1) (associativity): if (x , y) ∈ G(2) and (y, z) ∈ G(2), then so (x · y, z) ∈ G(2) and (x , y · z) ∈ G(2), and the relation, (x · y) · z = x · (y · z) is satisfied; (G2) (units): α ◦ ε = β ◦ ε = IdG0 and ε(α(x)) · x = x = x · ε(β(x)), (∀)x ∈ G; (G3) (inverses): for each x ∈ G we have α(x−1) = β(x), β(x−1) = α(x), x−1 · x = ε(β(x)) and x · x−1 = ε(α(x)). We sometimes use the notation x y instead of the product x · y . Whenever we write a product in a given groupoid, we are assuming that it is defined. The element ε(α(x)) (resp., ε(β(x))) is called the left unit (resp., right unit) of x; ε(G0) is called the unit set; x−1 is called the inverse of x . For a groupoid we use the notation (G,α,β , m,ε, i, G0) or (G, G0) or G. The functions α,β , m,ε, i are called structure functions. For each u ∈ G0, the set α−1(u) (resp., β−1(u)) is called α−fibre (resp., β−fibre) of G at u ∈ G0. For any u ∈ G0, the set G(u) := α−1(u)∩β−1(u) is a group under the restriction of the multiplication, called the isotropy group at u of the groupoid (G, G0). The map (α,β) : G → G0 × G0 defined by (α,β)(x) := (α(x),β(x)), (∀)x ∈ G is called the anchor map of G. A groupoid is transitive, if its anchor map is surjective. In particular, if (G,α,β , m,ε, i, G0) is a groupoid such that G0 ⊆ G and ε : G0 → G is the inclusion map, then (G,α,β , m, i, G0) is a Brandt groupoid, called G0−groupoid. Some elementary properties of groupoids are contained in the following proposition. Theorem 1. [8] In a groupoid (G, G0) the following assertions hold: (i) α(x y) = α(x) and β(x y) = β(y) for any (x , y) ∈ G(2); (ii) α ◦ i = β , β ◦ i = α and i ◦ i = IdG; (iii) i ◦ ε = ε and ε(u) · ε(u) = ε(u) for each u ∈ G0; (iv) i(x · y) = i(y) · i(x), for all (x , y) ∈ G(2); (v) ϕ : G(α(x))→ G(β(x)), ϕ(z) := x−1zx is an isomorphism of groups. (vi) if (G, G0) is transitive, then all isotropy groups are isomorphic. M. Ivan / Eur. J. Pure Appl. Math, 6 (2013), 469-484 471 Example 1. (i) A nonempty set G0 may be considered to be a groupoid over G0, called the null groupoid associated to G0. For this, we take α= β = ε = i = IdG0 and u · u= u for all u ∈ G0. (ii) A group G having e as unity has a structure of {e}−groupoid with respect to maps: α(x) = β(x) := e, G(2) = G × G, m(x , y) := x y, ε(e) := e and i(x) := x−1. Conversely, a groupoid with one unit (i.e., G0 = {e}) is a group. (iii) For the groupoids (G j ,α j ,β j , m j ,ε j , i j , G j,0), j = 1,2, one may construct the groupoid G1×G2 whose its structure functions are given by: α := α1×α2, β := β1×β2, ε := ε1×ε2, i := i1 × i2 and m((g1, g2), (g ′1, g ′2)) = (m1(g1, g ′1), m2(g2, g ′2)) for all (g, g ′1) ∈ G(2), (g2, g ′2) ∈ G(2). Then (G1 × G2,α, m,ε, i, G1,0 × G2,0) is a groupoid, called the direct product of (G1, G1,0) and (G2, G2,0). Definition 2 ([8]). Let (G,α,β , m,ε, i, G0) be a groupoid. (i) A pair (H, H0) of nonempty subsets where H ⊆ G and H0 ⊆ G0, is called subgroupoid of G, if: (1) α(H) = H0 and β(H) = H0; (2) H is closed under partially multiplication and inversion, that is: (a) (∀) x , y ∈ H such that (x , y) ∈ G(2) we have x · y ∈ H; (b) x−1 ∈ H, for all x ∈ H. (ii) A subgroupoid (H, H0) of (G, G0) is said to be wide, if H0 = G0. (iii) A wide subgroupoid (N , N0) of (G, G0) is called normal, if for all x ∈ G and a ∈ N we have x · a · x−1 ∈ N. Definition 3 ([9]). Let (G,α,β , G0) and (G′,α′,β ′, G′0) be two groupoids. (i) A morphism of groupoids or groupoid morphism from G into G′ is a pair ( f , f0) of maps f : G→ G′ and f0 : G0→ G′0 such that the following conditions hold: (i1) α′ ◦ f = f0 ◦α, β ′ ◦ f = f0 ◦ β; (i2) f (m(x , y)) = m′( f (x), f (y)) for all (x , y) ∈ G(2). (ii) If G0 = G′0 and f0 = IdG0 , we say that f is a G0−morphism of groupoids. (iii) A groupoid morphism ( f , f0) : (G, G0)→ (G′, G′0) such that f and f0 are bijective maps, is called isomorphism of groupoids. If ( f , f0) : (G, G0)→ (G′, G′0) is a groupoid morphism, then [8]: f ◦ ε = ε′ ◦ f0 and f ◦ i = i′ ◦ f . (1) M. Ivan / Eur. J. Pure Appl. Math, 6 (2013), 469-484 472 In the sequel we describe the notion of group-groupoid as algebraic structure (for definition see [4]). A group structure on a nonempty set is regarded as an universal algebra determined by a binary operation, an nullary operation and an unary operation. Let (G,α,β , m,ε, i, G0) be a groupoid. We suppose that on G is defined a group structure ω : G × G→ G, (x , y) 7→ω(x , y) := x ⊕ y . The unit element of the group G is denoted by e, that is ν : {λ} → G, λ 7→ ν(λ) := e (here {λ} is a singleton). The inverse of x ∈ G is denoted by x̄ , that is σ : G → G, x 7→ σ(x) := x̄ . Also, we suppose that on G0 is defined a group structure ω0 : G0× G0→ G0, (u, v) 7→ω0(u, v) := u⊕ v. The neutral element of the group G0 is denoted by e0, that is ν0 : {λ} → G0, λ 7→ ν0(λ) := e0. The inverse of u ∈ G0 is denoted by ū, that is σ0 : G0→ G0, u 7→ σ0(u) := ū. Definition 4 ([4]). A group-groupoid or G−groupoid, is a groupoid (G, G0) such that the following conditions hold: (i) (G,ω,ν ,σ) and (G0,ω0,ν0,σ0) are groups. (ii) The maps (ω,ω0) : (G× G, G0× G0)→ (G, G0), ν : {λ} → G and (σ,σ0) : (G, G0)→ (G, G0) are groupoid morphisms. We shall denote a group-groupoid by (G,α,β , m, i,ε,⊕, G0) or (G,α,β , m,⊕, G0). Theorem 2. If G,α,β , m,ε, i,⊕, G0) is a group-groupoid, then: (i) the multiplication m and the binary operation ω are compatible, that is: (x · y)⊕ (z · t) = (x ⊕ z) · (y ⊕ t), (∀)(x , y), (z, t) ∈ G(2); (2) (ii) the structure functions α,β : (G,⊕)→ (G0,⊕), ε : (G0,⊕)→ (G,⊕) and i : (G,⊕)→ (G,⊕) are morphisms of groups; (iii) the multiplication m and the unary operation σ are compatible, that is: σ(x · y) = σ(x) ·σ(y), (∀)(x , y) ∈ G(2). (3) Proof. By Definition 3, since (ω,ω0) is a groupoid morphism it follows that: (a) α ◦ω=ω0 ◦ (α×α) and β ◦ω=ω0 ◦ (β × β); (b) ω(mG×G((x , y), (z, t))) = mG(ω(x , z),ω(y, t)), (∀)(x , y), (z, t) ∈ G(2). (i) We have ω(mG×G((x , y), (z, t))) =ω(mG(x , y), mG(z, t)) =ω(x · y, z · t) = (x · y)⊕ (z · t) and mG(ω(x , z),ω(y, t)) = mG(x ⊕ z, y ⊕ t) = (x ⊕ z) · (y ⊕ t). Using (b) one obtains (x · y)⊕ (z · t) = (x ⊕ z) · (y ⊕ t), and (2) holds. M. Ivan / Eur. J. Pure Appl. Math, 6 (2013), 469-484 473 (ii) For each (x , y) ∈ G× G, we have α(ω(x , y)) = α(x ⊕ y) and ω0((α×α)(x , y)) =ω0(α(x),α(y)) = α(x)⊕α(y). According to the first equality (a), it follows α(x ⊕ y) = α(x)⊕ α(y), and α is a group morphism. Similarly, we prove that β is a group morphism. Since (ω,ω0) is a groupoid morphism, from (1) it follows. (c) ω ◦ (ε× ε) = ε ◦ω0 and i ◦ω=ω ◦ (i× i). For all u, v ∈ G0, we have ω((ε× ε)(u, v)) =ω(ε(u),ε(v)) = ε(u)⊕ ε(v) and ε(ω0(u, v)) = ε(u ⊕ v). From the first equality (c), it follows ε(u ⊕ v) = ε(u) ⊕ ε(v). Hence, ε is a group morphism. For all x , y ∈ G, we have i(ω(x , y)) = i(x ⊕ y) and ω(i(x), i(y)) = i(x)⊕ i(y). Using the second equality (c), it follows i(x ⊕ y) = i(x)⊕ i(y), and i is a group morphism. (iii) Since (σ,σ0) is a groupoid morphism, for all (x , y) ∈ G(2) we have σ(m(x , y)) = m(σ(x),σ(y)); i.e., σ(x · y) = σ(x) ·σ(y). Hence (3) holds. The relation (2) (resp., (3)) is called the interchange law between groupoid multiplication m and group operation ω (resp., σ). We say that the group-groupoid (G,α,β , m, i,ε,⊕, G0) is a commutative group-groupoid, if the groups G and G0 are commutative. Remark 1. (i) Let (G, G0) be a G−groupoid. For all x , y ∈ G, we have σ(x ⊕ y) = σ(y)⊕σ(x) and σ(σ(x)) = x; (ii) If (G, G0) is a commutative group-groupoid, then x ⊕ y = x̄ ⊕ ȳ , (∀) x , y ∈ G. Theorem 3. If (G,α,β , m,ε, i,⊕, G0) is a G−groupoid, then: e · y = y, (∀)y ∈ α−1(e0) and x · e = x , (∀)x ∈ β−1(e0); (4) x · (y ⊕ t) = x · y ⊕ t, (∀)(x , y) ∈ G(2) and t ∈ α−1(e0); (5) (x ⊕ z) · y = x · y ⊕ z, (∀)(x , y) ∈ G(2) and z ∈ β−1(e0). (6) M. Ivan / Eur. J. Pure Appl. Math, 6 (2013), 469-484 474 Proof. If y ∈ α−1(e0), then α(y) = e0. We have β(ε(e0)) = e0, since β ◦ ε = IdG0 . So (ε(e0), y) ∈ G(2). Using the condition (G2) from Definition 1, one obtains e · y = ε(e0) · y = ε(α(y)) · y = y . Hence the first relation of (4) holds. Similarly, we prove that the second relation of (4) hold. For to prove the relation (5) we apply the interchange law (2) and (4). Indeed, if in (2) we replace z with e, one obtains (x · y)⊕ (e · t) = (x ⊕ e) · (y ⊕ t), (∀)(x , y), (e, t) ∈ G(2). It follows (x · y)⊕ t = x · (y ⊕ t), since x ⊕ e = x ,β(e) = e0 and t ∈ α−1(e0). Hence, the relation (5) holds. Similarly, if in (2) we replace t with e, one obtains (x · y)⊕ (z · e) = (x ⊕ y) · (y ⊕ e), (∀)(x , y), (y, e) ∈ G(2). It follows (x · y)⊕ z = (x ⊕ z) · y , since y ⊕ e = y , α(e) = e0 and z ∈ β−1(e0). Hence, the relation (6) holds. Theorem 4. [3] If (G,α,β , m,ε, i,⊕, G0) is a G−groupoid, then: x · y = x ⊕ ε(β(x))⊕ y, (∀)(x , y) ∈ G(2); (7) x−1 = ε(α(x))⊕ x̄ ⊕ ε(β(x)), (∀)x ∈ G. (8) Proof. Let (x , y) ∈ G(2). Then β(x) = α(y). We have x · y = (x ⊕ ( ε(β(x))⊕ ε(β(x)))) · (e⊕ y), since ε(β(x))⊕ ε(β(x)) = e, x ⊕ e = x and e⊕ y = y . From the associativity of the law ⊕ and β(x) = α(y), one obtains x · y = ((x ⊕ ε(β(x)))⊕ ε(α(y))) · (e⊕ y). Applying the interchange law (2), the relations (4) and (G.2), we have x · y = ((x ⊕ (ε(β(x)) · e)⊕ (ε(α(y)) · y) ⇒ x · y = x ⊕ ε(β(x))⊕ y. Hence, the relation (7) holds. Applying the fact that α is a group morphism and the relation α ◦ ε = IdG0 , one obtains α(a) = α(ε(α(x)))⊕α( x̄)⊕α(ε(β(x))) = α(x)⊕α( x̄)⊕ β(x) = α(x ⊕ x̄)⊕ β(x) = β(x). From α(a) = β(x) it follows that the product x · a is defined. Applying the interchange law (2) and (4), we have x · a =(e⊕ x) · ((ε(α(x))⊕ x̄)⊕ ε(β(x))) = (e · (ε(α(x))⊕ x̄))⊕ (x · ε(β(x))) =ε(α(x))⊕ x̄ ⊕ x = ε(α(x)). Hence, x · a = ε(α(x)). Similarly, we verify that a · x = ε(β(x)). Then a = x−1 and the relation (8) holds. M. Ivan / Eur. J. Pure Appl. Math, 6 (2013), 469-484 475 Corollary 1. If (G,α,β , m,ε, i,⊕, G0) is a G−groupoid, then: x · y = x ⊕ y and x−1 = x̄ , (∀)x , y ∈ G(e0). (9) Proof. Let x , y ∈ G(e0). Then α(x) = α(y) = β(x) = β(y) = e0 and (x , y) ∈ G(2). Applying (7), we have x · y = x ⊕ y , since ε(β(x)) = ε(e0) = e. Hence, the first equality from (9) holds. Also, we have ε(α(x)) = ε(β(x)) = ε(e0) = e. Applying now (8), we have x−1 = x̄ . Hence, the second equality from (9) holds. 3. Category of Vector Space-groupoids Let (V,α,β , m,ε, i, V0) be a groupoid. We suppose that V (resp., V0) is a vector space over a field K . For the binary operation and unary operation in the group V (resp., V0) we will use the notations ω := + (resp., ω0 := +) and σ(x) := −x , x ∈ V (resp., σ0(u) := −u, u ∈ V0). The null vector of V (resp., V0) is e (resp., e0). The scalar multiplication ϕ : K×V → V (resp., ϕ0 : K × V0→ V0) is given by (k, x) 7→ ϕ(k, x) := kx (resp., (k, u) 7→ ϕ0(k, u) := ku). Consider the direct product (K×V, Id×α, Id×β , Id×m, Id×ε, Id× i, K×V0) of the null groupoid associated to K and groupoid (V, V0). Its set of composable elements is (K × V )(2) = {((k1, x), (k2, y)) ∈ (K × V )2 | k1 = k2, β(x) = α(y)}. The multiplication in K × V is given by (k, x) · (k, y) := (k, x · y), (∀)(x , y) ∈ V(2), k ∈ K . Definition 5. A vector space-groupoid or VS−groupoid, is a groupoid (V, V0) such that the following conditions hold: (5.1) (V,+,ϕ) and (V0,+,ϕ0) are vector spaces; (5.2) (V,α,β , m,ε, i,+, V0) is a commutative group-groupoid; (5.3) The pair (ϕ,ϕ0) : (K × V, K × V0)→ (V, V0) is a groupoid morphism. We shall denote a vector space-groupoid by (V,α,β , m, i,ε,+,ϕ, V0) or (V, V0). Theorem 5. If (V,α,β , m, i,ε,+,ϕ, V0) is a vector space-groupoid, then: (i) the multiplication m and the additive operation ω are compatible, that is: (x · y) + (z · t) = (x + z) · (y + t), (∀)(x , y), (z, t) ∈ V(2); (10) (ii) the structure functions α,β : (V,+)→ (V0,+), ε : (V0,+)→ (V,+) and i : (V,+)→ (V,+) are linear maps; M. Ivan / Eur. J. Pure Appl. Math, 6 (2013), 469-484 476 (iii) the multiplication m and the scalar multiplication ϕ are compatible, that is: k(x · y) = (kx) · (k y), (∀)(x , y) ∈ V(2) and k ∈ K; (11) (iv) the multiplication m and the unary operation σ are compatible, that is: −(x · y) = (−x) · (−y), (∀)(x , y) ∈ V(2). (12) Proof. (i) and (iv). Since (V,α,β , m,ε, i,+, V0) is a group-groupoid, it follows that the relation (10) holds and the structure functions α,β ,ε, i are morphisms from the corresponding additive groups. Also, Theorem 2(iii) it implies that the equality (12) is verified. From the fact that (ϕ,ϕ0) is a groupoid morphism, we have: (a) α ◦ϕ = ϕ0 ◦ (Id ×α) and β ◦ϕ = ϕ0 ◦ (Id × β); (b) ϕ((Id ×m)((k, x), (k, y))) = m(ϕ(k, x),ϕ(k, y)), (∀)(x , y) ∈ V(2), k ∈ K . (ii) and (iii). For each (k, x) ∈ K × V , we have α(ϕ(k, x)) = α(kx) and ϕ0((Id ×α)(k, x)) = ϕ0(k,α(x)) = kα(x). According to the first equality (a), it follows α(kx) = kα(x), and α is a linear map. Similarly, we prove that β is a linear map. We have ϕ((Id ×m)((k, x), (k, y))) = ϕ(k, m(x , y)) = km(x , y) = k(x · y) and m(ϕ(k, x),ϕ(k, y)) = m(kx , k y) = (kx) · (k y). Using (b) one obtains k(x · y) = (kx) · (k y), and (11) holds. Since (ϕ,ϕ0) is a groupoid morphism, from (1) it follows (c) ϕ ◦ (Id × ε) = ε ◦ϕ0 and i ◦ϕ = ϕ ◦ (Id × i). For all u ∈ V0 and k ∈ K , we have ϕ((Id × ε)(k, u)) = ϕ(k,ε(u)) = kε(u) and ε(ϕ0(k, u)) = ε(ku). From the first equality (c), it follows ε(ku) = kε(u). Hence, ε is a linear map. For all x ∈ V and k ∈ K , we have i(ϕ(k, x)) = i(kx) and ϕ(k, i(x)) = ki(x). Using the second equality (c), it follows i(kx) = ki(x), and i is a linear map. The relation (10) (resp., (11)) is called the interchange law between groupoid multiplica- tion m and scalar multiplication ω (resp., ϕ). The relation (12) is called the interchange law between groupoid multiplication m and group operation σ. From the Theorems 5, 1,3, 4 follows the following corollary. Corollary 2. Let (V,α,β , m, i,ε,+,ϕ, V0) be a VS−groupoid. Then: (i) The source and target α,β : V → V0 are surjective linear maps, and α(e) = β(e) = e0, α(−x) =−α(x) and β(−x) =−β(x), (∀) x ∈ V ; (ii) The inclusion map ε : V0→ V is an injective linear map, and ε(e0) = e, ε(−u) =−ε(u), (∀) u ∈ V0; M. Ivan / Eur. J. Pure Appl. Math, 6 (2013), 469-484 477 (iii) The inversion i : V → V is a linear automorphism, and i(e) = e, i(−x) =−i(x), (∀) x ∈ V ; (iv) The following assertions hold: e · y = y, (∀)y ∈ α−1(e0) and x · e = x , (∀)x ∈ β−1(e0); (13) x · (y + t) = x · y + t, (∀)(x , y) ∈ V(2) and t ∈ α−1(e0); (14) (x + z) · y = x · y + z, (∀)(x , y) ∈ V(2) and z ∈ β−1(e0); (15) x · y = x + y − ε(β(x)), (∀)(x , y) ∈ V(2); (16) x−1 = ε(α(x)) + ε(β(x))− x , (∀)x ∈ V. (17) Corollary 3. If (V,α,β , m, i,ε,+,ϕ, V0) is a VS−groupoid, then: x · y = x + y and x−1 =−x , (∀)x , y ∈ V (e0). (18) Proof. It follows immediately from (16) and (17). Theorem 6. Let (V,α,β , m,ε, i, V0) be a groupoid. If the following conditions are satisfied: (i) (V,+,ϕ) and (V0,+,ϕ0) are vector spaces; (ii) α,β : V → V0, ε : V0→ V and i : V → V are linear maps; (iii) the interchange law (10) between the operations m and ω holds, then (V,α,β , m,ε, i,+,ϕ, V0) is a vector space-groupoid. Proof. By hypothesis, the condition (5.1) from Definition 5 is verified. We prove now the condition (5.2) from Definition 5 is satisfied. The condition (i) from Definition 4 holds, since (V,ω,ν ,σ) and (V0,ω0,ν0,σ0) are commutative groups. (a) We prove that (ω,ω0) : (V × V, V0 × V0) → (V, V0) is a morphism of groupoids. Since α is a morphism of groups, it follows α(x + y) = α(x) + α(y), for all x , y ∈ V . Then α(ω(x , y)) =ω0(α(x),α(y)), and it follows α(ω(x , y)) =ω0((α×α)(x , y)); i.e., α ◦ω = ω0 ◦ (α × α). Similarly, we prove that β ◦ω = ω0 ◦ (β × β). Hence the condition (i1) from Definition 3(i) is satisfied. We suppose that the interchange law (10) holds. Then, for all (x , y) and (z, t) in G(2) we have (x · y) + (z · t) = (x + z) · (y + t). From the last equality it follows m(x , y)⊕m(z, t) =ω(x , z) ·ω(y, t) ⇒ ω(m(x , y), m(z, t)) = m(ω(x , z), (ω(y, t)). Then ω(mG×G((x , y), (z, t))) = m(ω(x , z), (ω(y, t)), and the condition (i2) from Defini- tion 3(i) holds. Hence, (ω,ω0) is a groupoid morphism. M. Ivan / Eur. J. Pure Appl. Math, 6 (2013), 469-484 478 (b) We prove that (ν ,ν0) is a morphism of groupoids (here {λ} is regarded as null groupoid with the structure functions α′0, β ′0, ε′0, i′0 and multiplication m′0). Since α and ε are group morphisms, we have α(e) = e0 and ε(e0) = e. From α(ν(λ)) = α(e) = e0 and ν0(λ) = e0, it follows α ◦ ν = ν0 ◦ Id. Similarly, we have β ◦ ν = ν0 ◦ Id. Also, we have ν(m′0(λ,λ)) = ν(λ) = e and m(ν(λ),ν(λ)) = e · e = ε(α(e)) · e = e. Then, ν(m′0(λ,λ)) = m(ν(λ),ν(λ)). Hence, (ν ,ν0) is a groupoid morphism. (c) We prove that (σ,σ0) is a groupoid morphism. Applying the fact that α is group mor- phism, we have α(σ(x)) = α(−x) =−α(x) and σ0(α(x)) =−α(x). Then α◦σ = σ0 ◦α. Similarly, we have β ◦σ = σ0 ◦ β . We shall prove that: (c1) −x · y = (−x) · (−y), (∀) (x , y) ∈ V(2). From (x , y) ∈ V(2) we have β(x) = α(y). Then β(−x) = α(−y). Therefore (−x ,−y) ∈ V(2). Using now (10) one obtains (c2) (x · y) + ((−x) · (−y)) = (x + (−x)) · (y + (−y)) and (c3) ((−x) · (−y)) + (x · y) = ((−x) + x) · ((−y) + y). Since a+ (−a) = (−a) + a = e, and e · e = e, from (c2) and (c3), we have (c4) (x · y) + ((−x) · (−y)) = e and ((−x) · (−y)) + (x · y) = e. From (c4) one obtains that the equality (c1) holds. The relation (c1) is equivalently with σ(x · y) = si gma(x) · σ(y). Then σ(m(x , y)) = m(σ(x),σ(y)). Hence, (σ,σ0) is a groupoid morphism. Therefore, (V,α,β , m,ε, i,+, V0) is a commutative group-groupoid and the condition (5.2) from Definition 5 holds. We shall prove that (ϕ,ϕ0) : (K × V, K × V0)→ (V, V0) is a groupoid morphism. Applying the fact that α is a linear map, for all x ∈ V and k ∈ K we have α(ϕ(k, x)) = α(kx) = kα(x) and ϕ0((Id ×α)(k, x)) = ϕ0(k,α(x)) = kα(x). Then α ◦ϕ = ϕ0 ◦ (Id ×α). Similarly, we have β ◦ϕ = ϕ0 ◦ (Id × β). We consider x , y ∈ V such that (x , y) ∈ V(2). We have also (kx , k y) ∈ V2. Indeed, using the linearity of α and β , from β(x) = α(y) follows β(kx) = α(k y). Applying now the relation (16), linearity of ε and β and the fact that V is a vector space, we have k(x · y) = k(x + y − ε(β(x))) = kx + k y − kε(β(x)) and (kx) · (k y) = kx + k y − ε(β(kx))) = kx + k y − kε(β(x)). Then k(x · y) = (kx) · (k y), for all (x , y) ∈ V(2) and k ∈ K; i.e. the interchange law (11) holds. From (11) it follows ϕ(k, m(x , y)) = ϕ(k, x) ·ϕ(k, y) ⇒ ϕ((Id ×m)((k, x), (k, y))) = m(ϕ(k, x),ϕ(k, y)). Therefore, (ϕ,ϕ0) is a groupoid morphism. Hence, (V,α,β , m,ε, i,+,ϕ, V0) is vector space-groupoid. M. Ivan / Eur. J. Pure Appl. Math, 6 (2013), 469-484 479 According to Theorems 5 and 6, we can give another definition for the notion of vector space-groupoid (this is equivalent with Definition 5). Definition 6. A vector space-groupoid is a groupoid (V,α,β , m,ε, i, V0) such that the following conditions are satisfied: (i) (V,+,ϕ) and (V0,+,ϕ0) are vector spaces; (ii) α,β : V → V0, ε : V0→ V and i : V → V are linear maps; (iii) the interchange law (10) between the operations m and ω holds. If in Definition 6, we consider V0 ⊆ V and ε : V0 → V is the inclusion map, then (V,α,β , m, i,+,ϕ, V0) is a vector space-groupoid. In this case, we will say that (V, V0) is a vector space−V0−groupoid. Example 2. (i) Let (V,+,ϕ) be a vector space. Then V has a structure of null groupoid over V (see Example 1(i)). We have that V0 = V and α,β ,ε, i are linear maps. It is easy to verify that the interchange law (10) holds. Then V is a vector space-groupoid, called the null vector space-groupoid associated to V . (ii) Let (V,+,ϕ) be a vector space having {e} as null vector. The set V is a {e}−groupoid (see Example 1(ii)). In this case, m = +. We have that V0 := {e} is a vector subspace in V and α, β , ε and i are linear maps. The relation (10) holds. Indeed, for x , y, z, t ∈ V we have (x + y) + (z+ t) = (x + z) + (y + t), since the addition operation is associative and commutative. Hence (V,α,β , m,ε, i,+,ϕ, {e}) is a vector space-groupoid called the vector space-groupoid with a single unit associated to V . Therefore, each vector space V has a structure of vector space−{e}−groupoid. Definition 7. Let (V,α,β , m,ε, i,+,ϕ, {e}) be a vector space-groupoid. (i) By a vector space-subgroupoid (resp., vector space-wide subgroupoid or vector space −V0−subgroupoid) of (V, V0), we mean a subgroupoid (resp., wide subgroupoid) (W, W0) of the groupoid (V, V0) with the property that W and W0 are vector subspaces in V and V0, respectively. (ii) A vector space-subgroupoid (N , V0) of (V, V0) is called vector space-normal subgroupoid, if (N , V0) is a normal subgroupoid of the groupoid (V, V0). According to the Definition 6, if (W, W0) is a vector space-subgroupoid of (V, V0), then the pair (W, W0) endowed with the restrictions of the functions α,β , i and + to W , the restriction of ε to W0 and the restriction of m to W(2), is a vector space-groupoid, denoted by (W, W0). Theorem 7. Let (V,α,β , m,ε, i,+,ϕ, V0) be a vector space-groupoid. Then: M. Ivan / Eur. J. Pure Appl. Math, 6 (2013), 469-484 480 (i) The fibres α−1(e0) and β−1(e0) are vector subspaces in V . (ii) The isotropy group V (e0) is a vector space−{e0}−subgroupoid of V . (iii) ε(V0) is a vector space −normal subgroupoid of V . (iv) Is(V ) := {x ∈ V | α(x) = β(x)} is a vector space −normal subgroupoid of V . Proof. (i) For all x , y ∈ α−1(e0) and k ∈ K , we have α(x − y) = α(x)−α(y) = e0 and α(kx) = kα(x) = ke0 = e0. Then x − y, kx ∈ α−1(e0). Hence α−1(e0) is a vector subspace. Similarly, we prove that β−1(e0) is a vector subspace. (ii) V (e0) is a vector subspace, since V (e0) = α−1(e0)∩ β−1(e0). Also, V (e0) is a {e0}−subgroupoid. Then, V (e0) is a vector space−{e0}−subgroupoid of V . For to prove the following assertions, we apply the Theorem 1. (iii) For x , y ∈ ε(V0) there exist u, v ∈ V0 such that ε(u) = x and ε(v) = y . It follows β(x) = β(ε(u)) = u and α(y) = α(ε(v)) = v. We suppose that the product x · y is defined. From β(x) = α(y) it follows x = y . Then x · y = ε(u) · ε(u) = ε(u) ∈ ε(V0). Also, for x ∈ ε(V0), we have x−1 = i(x) = i(ε(u)) = ε(u) ∈ ε(V0). Let now a ∈ V and x ∈ ε(V0) such that a · x · a−1 is defined. From x = ε(u) and β(a) = α(x) it follows β(a) = α(ε(u)) = u and x = ε(β(a)). Then a · x · a−1 = (a · ε(β(a))) · a−1 = a · a−1 = ε(α(a)) ∈ ε(V0). Hence, ε(V0) is a normal subgroupoid. Also, ε(V0) is a vector subspace in V , since ε : V0→ V is a linear map. Therefore, ε(V0) is a vector space-normal subgroupoid. (iv) Clearly, α(Is(V )) = β(Is(V )) = V0. Let x , y ∈ Is(V ) with (x , y) ∈ V(2). Then α(x) = β(x) = α(y) = β(y). We have α(x y) = β(x y) and α(x−1) = β(x−1). It follows that x y, x−1 ∈ Is(V ). Let now a ∈ V and x ∈ Is(V ) such that a · x · a−1 is defined. From α(a · x · a−1) = α(a) and β(a · x · a−1) = β(a−1) = α(a) it follows α(a · x ·a−1) = β(a · x ·a−1). Then a · x ·a−1 ∈ Is(V ) and Is(V ) is a normal subgroupoid. Using the linearity of α and β , we have α(x − y) = β(x − y) and α(kx) = β(kx) for all x , y ∈ Is(V ) and k ∈ K . Therefore, Is(V ) is a vector subspace. Hence, Is(V ) is a vector space− normal subgroupoid. The group-subgroupoid Is(V ) is the union of all isotropy groups of V and it is called the isotropy bundle of the vector space-groupoid (V, V0). Example 3. Let a ∈ R, a 6= 1. Consider the vector spaces groups V := R3 and V0 := R. For (V, V0), we define the structure functions α,β : R3→ R, ε : R→ R3 and M. Ivan / Eur. J. Pure Appl. Math, 6 (2013), 469-484 481 i : R3 → R3 as follows: α(x1, x2, x3) := ax1 + x2, β(x1, x2, x3) := x1 + x2, ε(x2) := (0, x2, 0) and i(x1, x2, x3) := (−x1, (a+ 1)x1+ x2,−x3), for all x1, x2, x3 ∈ R. Let V(2) := {((x1, x2, x3), (y1, y2, y3)) ∈ R3×R3 | y2 = x1+ x2− a y1} be the set of compos- able pairs. The multiplication m : V(2)→ V is given by: (x1, x2, x3) · (y1, y2, y3) := (x1+ y1, x2− a y1, x3+ y3), if y2 = x1+ x2− a y1. It is easy to check that the above structure functions determine on V a structure of a groupoid over V0. Also, the maps α,β ,ε and i are linear maps. Therefore, the conditions (i) and (ii) from the Definition 6 hold. Let x , y, z, t ∈ R3 such that x · y and z · t are defined. Then x = (x1, x2, x3), y = (y1, y2, y3), z = (z1, z2, z3), t = (t1, t2, t3) such that y2 = x1 + x1 − a y1 and t2 = z1 + z2 − at1. We have x · y = (x1+ y1, x2− a y1, x3+ y3), z · t = (z1+ t1, z2− at1, z3+ t3). Then (x · y) + (z · t) = (x1+ y1+ z1+ t1, x2− a y1+ z2− at1, x3+ y3+ z3+ t3) and (x + z) · (y + t) = (x1+ z1+ y1+ t1, x2+ z2− a(y1+ t1), x3+ z3+ y3+ t3). Hence, (x · y) + (z · t) = (x + y) · (z + t) and the interchange law (10) holds. Therefore, (R3,α,β , m,ε, i,+,ϕ,R) is a vector space-groupoid. We have ε(V0) = {(0, u, 0)|u ∈ R} and Is(V ) = {(0, u, v)|u, v ∈ R} are vector space-normal subgroupoid. The isotropy group at e0 = 0 is V (0) = {(0,0, v)|v ∈ R}. Let us we consider the Euclidean space R3 with the Cartesian coordinate system Ox1 x2 x3. The α−fibres α−1(u) for u ∈ R are represented by parallel planes of equation x1+2x2−u= 0. Also, the β−fibres β−1(v) for v ∈ R are represented by parallel planes of equation x1+ x2− v = 0. Let be the points A1, A2, A3, A4 associated to elements ε(β(x)), x , x · y, y ∈ V , for β(x) = α(y). We have A1(0, b1 + b2, 0), A2(b1, b2, b3), A3(b1 + c1, b2 − ac1, b3 + c3) and A4(c1, b1+ b2− ac1, c3). Then: the simple quadrilateral A1A2A3A4 is a parallelogram. Indeed, the straight line through A1 and A4 has the equation: x1 c1 = x2−(b1+b2) −ac1 = x3 c3 and the distance from A1 and A4 is d(A1, A4) = p (1+ a2)c2 1 + c2 3 . Also, the straight line through A2 and A3 has the equation: x1−b1 c1 = x2−b2 −ac1 = x3−b3 c3 and the distance from A2 and A3 is d(A2, A3) = p (1+ a2)c2 1 + c2 3 . Let be the points B1, B2, B3, B4 associated to ε(α(x)), x ,ε(β(x)), x−1 ∈ V . We have B1(0, ab1+ b2, 0), B2(b1, b2, b3), B3(0, b1+ b2, 0) and B4(−b1, (a+1)b1+ b2,−b3). Then: the simple quadrilateral B1B2B3B4 is a parallelogram. Indeed, the straight line through B1 and B2 has the equation: x1 b1 = x2−(ab1+b2) −ab1 = x3 b3 and d(B1, B2) = p (1+ a2)b2 1 + b2 3. Also, the straight line through B3 and B4 has the equation: x1 −b1 = x2−(b1+b2) ab1 = x3 −b3 and d(B3, B4) = d(B1, B2). Definition 8. Let (Vj ,α j ,β j , m j ,ε j , i j ,+ j ,ϕ j , Vj,0), j = 1, 2 be two vector space-groupoids. A groupoid morphism ( f , f0) : (V1, V1,0)→ (V2, V2,0) with property that f : V1→ V2 and M. Ivan / Eur. J. Pure Appl. Math, 6 (2013), 469-484 482 f0 : V1,0 → V2,0 are linear maps, is called vector space-groupoid morphism or morphism of vector space-groupoids. If V2,0 = V1,0 and f0 = IdV1,0 , then we say that ( f , IdV1,0 ) : (V1, V1,0) → (V2, V1,0) is a V1,0−morphism of vector space-groupoids. It is denoted by f : V1→ V2. The category of vector space-groupoids, denoted by VSGpd, has its objects all vector space-groupoids (V, V0) and as morphisms from (V, V0) to (V ′, V ′0) the set of all morphisms of vector space-groupoids. Finally we will present the concept of vector groupoid defined by V. Popuţa and Gh. Ivan [13, 14]. Definition 9 ([13]). By vector groupoid, we mean a groupoid (V,α,β , m,ε, i, V0) which verifies the following conditions: (9.1) V and V0 are vector spaces; (9.2) α,β : V → V0 are linear maps; (9.3) the inclusion ε : V0 → V and the inversion i : V → V are linear maps and the following condition is verified: (9.3.1) x + i(x) = ε(α(x)) + ε(β(x)) for all x ∈ V; (9.4) the multiplication m : V(2)→ V satisfy the following relations: (9.4.1) x · (y + z− ε(β(x))) = x · y + x · z− x, (∀) x , y, z ∈ V such that α(y) = β(x) = α(z); (9.4.2) x · (k y + (1− k)ε(β(x))) = k(x · y) + (1− k)x, (∀) (x , y) ∈ V(2); (9.4.3) (y + z− ε(α(x))) · x = y · x + z · x − x, (∀) x , y, z ∈ V such that α(x) = β(y) = β(z); (9.4.4) (k y + (1− k)ε(α(x))) · x = k(y · x) + (1− k)x, (∀) (y, x) ∈ V(2). Theorem 8. Each vector space-groupoid is a vector groupoid in the sense of Definition 9. Proof. We suppose that (V,α,β , m,ε, i,+,ϕ, V0) is a vector space-groupoid. From Defini- tion 6 and (17) it follows that the conditions (9.1)− (9.3) are satisfied. Let x , y, z ∈ V such that α(y) = β(x) = α(z). Denote t := y + z − ε(β(x)). Using the linearity of α, we have α(t) = α(y) +α(z)−α(ε(β(x))) = β(x) and (x , t) ∈ V(2). Applying (16), we have x · t = x + t − ε(β(x)) = x + y + z− 2ε(β(x)) and x · y + x · z− x = (x + y − ε(β(x))) + (x + z− ε(β(x)))− x = x + y + z− 2ε(β(x)). Then, x · t = x · y + x · z− x . Hence, the relation (9.4.1) holds. Let y, x) ∈ V(2). Denote v := k y + (1 − k)ε(α(x)). Using the linearity of β , we have β(v) = kβ(y) + (1− k)β(ε(α(x))) = α(x), since β(y) = α(x). Then (v, x) ∈ V(2). REFERENCES 483 Applying (16), we have v · x = v + x − ε(β(v)) = x + k y − kε(α(x)) and k(y · x) + (1− k)x = k(y + x − ε(β(y))) + (1− k)x = x + k y − kε(α(x)). Then, v · x = k(y · x) + (1− k)x . Hence, (9.4.4) holds. Similarly, we prove that (9.4.2) and (9.4.3) are verified. Therefore, (V, V0) is a vector groupoid. References [1] H. Brandt. 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