A METHOD FOR COMPUTING OF GROUP

Analele Universit˘ a¸tii Oradea Fasc. Matematica, Tom XXV (2018), Issue No. 1, 25–32

A METHOD FOR COMPUTING OF GROUP-SUBGROUPOIDS OF FINITE GROUP-GROUPOIDS MIHAI IVAN1 AND MARIAN DEGERATU2

Abstract. In this paper we present an algorithm for finding of group-subgroupoids of a finite group-groupoid.

1. Introduction The concept of groupoid was introduced by H. Brandt [Math. Ann., 96(1926), 360-366]. The notion of group-groupoid was defined by R. Brown and Spencer [2]. In algebraically context the definition of group-groupoid is given in [6]. A group-groupoid is an algebraic structure which combines the concepts of group and groupoid such that these are compatible. The groupoids and group-groupoids have important applications in various areas of science [7, 1, 2, 3]. The paper is organized as follows. In Section 2 we present some useful definitions about of group-groupoids. In Section 3 we give a practical method for determine the groupsubgroupoids of a finite group-groupoid. Using the program BGroidAP 2 ([5]) for two finite group-groupoids, its group-subgroupoids are computed. 2. Group-groupoids The concept of group-groupoid as algebraic structure is described in [6]. For more details concerning the properties of groupoids and group-groupoids, see [7, 4, 2]. Let (G, G0 ) be a pair of nonempty sets with G0 ⊆ G. A group structure on the set G is defined by the operation ⊕ : G × G → G, (x, y) → x ⊕ y and G0 is a subgroup of (G, ⊕). The unit element of G and the inverse of x ∈ G are e and x ¯, respectively. We suppose that the pair of nonempty sets (G, G0 ) is endowed with two surjective maps α, β : G → G0 , a map ι : G → G and a partially multiplication : G(2) := {(x, y) ∈ G × G | β (x) = α (y)} → G, (x, y) → x y. Definition 2.1. ([6]) A group-groupoid is a groupoid (G, α, β, , ι, G0 ) such that the following conditions are satisfied: (i) (G, ⊕) and (G0 , ⊕) are groups; (ii) α, β : (G, ⊕) → (G0 , ⊕), and ι : (G, ⊕) → (G, ⊕) are morphisms of groups; (iii) the multiplication and the operation ⊕ are compatible, that is: (x y) ⊕ (z t) = (x ⊕ z) (y ⊕ t), (∀)(x, y), (z, t) ∈ G(2) .

(2.1)

The relation (2.1) is called the interchange law between the groupoid multiplication and the group operation ⊕. A group-groupoid is denoted by (G, α, β, , ι, ⊕, G0 ). 2010 Mathematics Subject Classification. 20L13, 68W10. Key words and phrases. groupoid, group-groupoid, group- subgroupoid. 25

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MIHAI IVAN AND MARIAN DEGERATU

We recall that (G, α, β, , ι, G0 ) has a structure of groupoid [7, 4], if the following properties are satisfied : (G1) (associativity): (x y) z = x (y z) for all (x, y) ∈ G(2) and (y, z) ∈ G(2) ; (G2) (units): ∀x ∈ G ⇒ (α(x), x), (x, β(x)) ∈ G(2) and α(x) x = x β(x) = x; (G3) (inverses): ∀x ∈ G ⇒ (x, ι(x), (ι(x), x) ∈ G(2) and we have ι(x) x = β(x) and x ι(x) = α(x). A group-groupoid (G, α, β, , ι, ⊕, G0 ) is also called group-groupoid over G0 with the structure functions α (source), β (target), (multiplication), ι (inversion) and ⊕ (addition). G(2) is the set of composable pairs and G0 is the set of units of G. Definition 2.2. ([4, 6]) Let (G, α, β, , ι, ⊕, G0 ) be a group-groupoid. A pair of nonempty subsets (H, H0 ) where H ⊆ G and H0 ⊆ G0 , is called group-subgroupoid of the groupgroupoid (G, G0 ), if the following conditions hold: (1) (H, H0 ) is a subgroupoid of the groupoid (G, G0 ), that is: (i) α(H) = H0 and β(H) = H0 ; (ii) ∀ x, y ∈ H such that (x, y) ∈ G(2) =⇒ x y ∈ H; (iii) ∀ x ∈ H =⇒ ι(x) ∈ H. (2) H and H0 are subgroups in G and G0 , respectively. Proposition 2.1. ([6]) Let (G, α, β, , ι, ⊕, G0 ) be a group-groupoid. Then: (i) G0 and G are group-subgroupoids of G. (ii) The isotropy group G(e0 ) = {x ∈ G|α(x) = β(x) = e0 } and isotropy bundle Is(G) := {x ∈ G|α(x) = β(x)} are group-subgroupoids of G. Example 2.2. The pair group-groupoid G 2 associated to G. Let (G, ⊕) be a commutative group. Consider the sets G := G × G and G0 := {(x, x) ∈ G × G|∀x ∈ G}. The pair (G, G0 ) has a structure of groupoid by taking α, β : G −→ G0 , ι : G −→ G and : G(2) := {(x, y), (y, z) ∈ G × G|∀x, y, z ∈ G} −→ G as follows: α(x, y) := (x, x); β(x, y) := (y, y); ι(x, y) := (y, x); (x, y) (y, z) := (x, z). It is easy to check that (G = G 2 , α, β, , ι, G0 ) is a groupoid. G = G × G is a group endowed with operation (x1 , x2 ) ⊕ (y1 , y2 ) := (x1 ⊕ y1 , x2 ⊕ y2 ), ∀ x1 , x2 , y1 , y2 ∈ G. Also, (G0 , ⊕) is a subgroup of (G, ⊕). It easy to prove that α, β, ι are group morphisms. Let x = (x1 , x2 ), y = (y1 , y2 ), z = (z1 , z2 ), t = (t1 , t2 ) from G × G such that β(x) = α(y), β(z) = α(t). Then y1 = x2 , t1 = z2 . It follows y = (x2 , y2 ), t = (z2 , t2 ), x y = (x1 , x2 ) (x2 , y2 ) = (x1 , y2 ) and z t = (z1 , z2 ) (z2 , t2 ) = (z1 , t2 ). We have (x y) ⊕ (z t) = (x1 , y2 ) ⊕ (z1 , t2 ) = (x1 ⊕ z1 , y2 ⊕ t2 ) and (x ⊕ z) (y ⊕ t) = (x1 ⊕ z1 , x2 ⊕ z2 ) (x2 ⊕ z2 , y2 ⊕ t2 ) = (x1 ⊕ z1 , y2 ⊕ t2 ). Then, (x y) ⊕ (z t) = (x ⊕ z) (y ⊕ t) and so (2.1) holds. Hence (G = G 2 , α, β, , ι, +, G0 ) is a group-groupoid, called the pair group-groupoid associated to G and denoted by G 2 . Remark 2.1. According to Proposition 2.1, it follows that the group-groupoid G 2 contains the following three group-subgroupoids: {(0)}, G0 = Is(G 2 ) = {(x, x) ∈ G × G | ∀x ∈ G} and G 2 . In particular, if n is a positive integer (n > 1), then Z2n is a group-groupoid, called the pair group-groupoid associated to group (Zn , +) of integers modulo n. Example 2.3. The group-groupoid R(d). Let (R, +, ·, 0, 1) be a commutative ring with identity. Let d ∈ R \ {0} be a divisor of zero such that 2d − 1 is invertible and d2 = d. Define the map fd : R → R given by fd (x) = dx, ∀x ∈ R.

A METHOD FOR COMPUTING OF GROUP-SUBGROUPOIDS OF FINITE GROUP-GROUPOIDS

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We consider G := R and G0 := Imfd . We have that G is a commutative group under addition and G0 ⊆ G is a subgroup. The functions αd , βd : G −→ G0 , ιd : G −→ G and multiplication d : G(2) := {(x, y) ∈ R × R | dx = dy} −→ G are given by: αd (x) := dx; βd (x) := dx; ιd (x) := 2dx − x; x d y := x + y − dy. Let x, y, z ∈ G with (x, y) ∈ G(2) and (y, z) ∈ G(2) . Then dx = dy = dz. We have (x d y) d z = (x d y) + z − dz = x + y + z − dy − dz and x d (y d z) = x + (y d z) − dx = x + (y d z) − dy = x + y + z − dy − dz. It follows (x d y) d z = x d (y d z and so the multiplication is associative. For x ∈ G, we have (αd (x), x) ∈ G(2) , since βd ((αd )(x)) = dαd (x) = d2 x = dx = αd (x). Also, αd (x) d x = dx d x = dx+x−dx = x. Similarly, (x, βd (x)) ∈ G(2) and x d βd (x) = x. For x ∈ G =⇒ (x, ιd (x)) ∈ G(2) , since αd (ιd (x)) = d(2dx − x) = dx = βd (x). Also, x d ιd (x) = x d (2dx−x) = dx = αd (x). Similarly, (ιd (x), x) ∈ G(2) and ιd (x) d x = βd (x). Therefore, the conditions (G1) − (G3) hold and G has a groupoid structure. It is easy to verify that αd , βd : (G, +) → (G0 , +) and ιd : (G, +) → (G, +) are group morphisms. Let x, y, z, t ∈ G with βd (x) = αd (y) and βd (z) = αd (t). Then dx = dy and dz = dt. We have (x d y) + (z d t) = (x + y − dx) + (z + t − dz) = x + y + z + t − dx − dz and (x + z) d (y + t) = (x + z) + (y + t) − d(x + z) = x + y + z + t − dx − dz. Then, (x d y) + (z d t) = (x + z) d (y + t) and so (2.1) holds. Hence G is a group-groupoid, called the group-groupoid of type d associated to ring R and denoted by (R(d), αd , βd , d , ιd , +, G0 ) or R(d). Remark 2.2. According to Proposition 2.1, it follows that the group-groupoid R2 (d) contains the following three group-subgroupoids: {(0)}, G0 = Is(R(d)) and R(d). If R := Zn , where n is a positive integer (n > 1) then one obtains the group-groupoid of type d associated to ring Zn of integers modulo n, denoted by Zn (d). In particular, Z10 (5), Z12 (4), Z14 (7), Z18 (9) and Z20 (5) are group-groupoids. 3. Computational method for finding of group-subgroupoids of a finite group-groupoid A finite groupoid (G, G0 ) such that |G| = n and |G0 | = m, 1 ≤ m ≤ n (here, |G| denotes the cardinality of G), is called finite groupoid of type (n; m). Each finite groupoid of type (n; 1) is a group. Denote a finite groupoid of type (n; m) by G(n;m) . Let us give an algorithm to find the group-subgroupoids of the group-groupoid (G, α, β, , ι, ⊕, G0 ) of type (n; m), where G = {a1 , a2 , · · · , am , am+1 , · · · , an } and G0 = {a1 , a2 , · · · , am }. This algorithm is constituted by the following stages. Stage 1.We determine the set Hsgroid (G) of subgroupoids of the groupoid (G, G0 ). For this we apply the program BGroidAP 2, given in [5]. This stage is realized in the following steps: Step 1.1. We introduce the initial data: n = |G|, m = |G0 |; the functions α, β, ι and given by its tables of structure. Step 1.2. Test if the universal algebra (G, α, β, , ι, G0 ) is a groupoid. Step 1.3. Determine the subgroupoids of (G, G0 ). The correspondence between the initial data and input data is given as follows: G = {a1 , a2 , . . . , . . . , am , am+1 , . . . , an } ←→ {1, 2, . . . , m, m + 1, . . . , n} Initial data ←→ Input data |G| = n ←→ n |G0 | = m ←→ m

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ak α(ak ) β(ak ) ι(ak )

a1 · · · a1 · · · a1 · · · a1 · · ·

a1 a1 ··· aj ··· an

· · · an · · · α(an ) 1 · · · m ul (m + 1) · · · ul (n) ←→ 1 · · · m ur (m + 1) · · · ur (n) · · · β(an ) 1 · · · m inv(m + 1) · · · inv(n) · · · ι(an )

am am+1 am α(am+1 ) am β(am+1 ) am ι(am+1 )

···

···

ak

an

aj ak

a11 a21 ←→ · · · aj1 ··· an1

··· ··· ··· ··· ··· ···

a1k a2k ··· ajk ··· ank

··· ··· ··· ··· ··· ···

a1n a2n ··· ajn ··· ann .

The mappings ul and ur are defined by: ul (m + j) = k, where j ∈ {1, 2, . . . , n − m} and k ∈ {1, 2, . . . , r} ←→ α(am+j ) = ak and ur (m + s) = t, where s ∈ {1, 2, . . . , n − m} and t ∈ {1, 2, . . . , r} ←→ β(am+s ) = at If the product aj ak is not defined, then this fact is represented by 0 in the table of input data. Using the correspondence between output data and final data one obtain all subgroupoids. Stage 2. We determine the set Ksgr (G) of subgroups of the group (G, ⊕). We apply the program BGroidAP 2. The group (G, ⊕) is regarded as a groupoid (G, α0 , β0 , ⊕, ι0 , {e0 } of type (n; 1), where α0 , β0 : G → {e0 } and ι0 : G → G are defined by α0 (x) = β0 (x) = e0 , ι0 (x) = x, ∀x ∈ G. Stage 3. Compute the intersection of the sets Hsgroid (G) and Ksgr (G) and obtain the set Sgrsgroid (G) of group-subgroupoids of the group-groupoid (G, G0 ). We illustrate the applicability of this method by the following examples. Example 3.1. Determination of group-subgroupoids of the pair group-groupoid Z23 . Let be the group-groupoid (Z23 , G0 ) of type (9; 3), see Example 2.1. We have Z32 = {p1 = (0, 0), p2 = (1, 1), p3 = (2, 2), p4 = (0, 1), p5 = (0, 2), p6 = (1, 0), p7 = (1, 2), p8 = (2, 0), p9 = (2, 1)} and G0 = {p1 = (0, 0), p2 = (1, 1), p3 = (2, 2)}. Stage 1. We determine the set Hsgroid (Z23 ) of subgroupoids of groupoid (Z23 , α, β, , ι, G0 ). For this we apply the program BGroidAP 2, given in [5]. The correspondence between the initial data and input data are the following: Z23 = {p1 , p2 , p3 , p4 , p5 , p6 , p7 , p8 , p9 } Initial data |Z23 | = 9 |G0 | = 3 pj α(pj ) β(pj ) ι(pj )

p1 p1 p1 p1

p2 p2 p2 p2

p3 p3 p3 p3

p4 p1 p2 p6

p5 p1 p3 p8

p6 p2 p1 p4

p7 p2 p3 p9

p8 p3 p1 p5

←→ {1, 2, 3, 4, 5, 6, 7, 8, 9} ←→ Input data ←→ 9 ←→ 3

p9 p3 1 ←→ 1 p2 1 p7

2 2 2

3 3 3

1 2 6

1 3 8

2 1 4

2 3 9

3 1 5

3 2 7


p1 p2 p3 p4 p5 p6 p7 p8 p9

p1 p1

p2

p3

p4 p4

p5 p5

p2

p6

p7

p6

p7

p1

p5

p3 p4 p5 p6

p2

p7

p9

p3

p7 p8 p9

p8

p8

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p9

p8

p9

p1

p4

p6

p2

p3

1 0 0 0 ←→ 0 6 0 8 0

0 2 0 4 0 0 0 0 9

0 0 3 0 5 0 7 0 0

4 0 0 0 0 2 0 9 0

5 0 0 0 0 7 0 3 0

0 6 0 1 0 0 0 0 8

0 7 0 5 0 0 0 0 3

0 0 8 0 1 0 6 0 0

0 0 9 0 4 0 2 0 0

The program BGroidAP 2 for these input data, confirm groupoid structure of Z23 and displays the 14 subgroupoids. The subgroupoids of Z23 containing {p1 } and whose order k is a divisor of n = 9, are given in the following correspondence: Output program {1}, {1, 2, 3} {1, 2, 3, 4, 5, 6, 7, 8, 9}

Subgroupoids 1 2 H(1;1) = {p1 }, H(3;3) = {p1 , p2 , p3 } = G0 3 2 H(9;3) = Z3

Stage 2. We determine the set Ksgr (Z23 ) of subgroups of the group ((Z23 , +). The unit set is {p1 = (0, 0)}. We apply the program BGroidAP 2. The input data for the groupoid (Z23 , +) are as follows:

9 1 1 1 1

1 1 3

1 1 2

1 1 5

1 1 4

1 1 8

1 1 9

1 1 6

1 1 7

1 2 3 4 5 6 7 8 9

2 3 1 7 6 9 8 4 5

3 1 2 8 9 5 4 7 6

4 7 8 5 1 2 6 9 3

5 6 9 1 4 7 2 3 8

6 9 5 2 7 8 3 1 4

7 8 4 6 2 3 9 5 1

8 4 7 9 3 1 5 6 2

9 5 6 3 8 4 1 2 7

The program BGroidAP 2 for these input data corresponding to group (Z23 , +) confirm the group structure (groupoid structure of type (9; 1)) and displays the 6 subgroups. The subgroups of Z23 are given in the following correspondence: Output program {1} {1, 2, 3}, {1, 4, 5} {1, 6, 8}, {1, 7, 9} {1, 2, 3, 4, 5, 6, 7, 8, 9}

Subgroups 1 K(1;1) = {p1 } 2 3 K(3;1) = {p1 , p2 , p3 } = G0 , K(3;1) = {p1 , p4 , p5 } 4 5 K(3;1) = {p1 , p6 , p8 }, K(3;1) = {p1 , p7 , p9 } 6 K(9;1) = Z23

Stage 3. Computing the intersection of the sets Hsgroid (Z23 ) and Ksgr (Z23 ) and one 1 2 3 obtains the set Sgrsgroid (Z23 ) = {H(1;1) , H(3;3) , H(9;3) }. 2 We conclude that the group-groupoid Z3 has only three group-subgroupoids, namely: H1 = {(0, 0)}, H2 = {(0, 0), (1, 1), (2, 2)} and H3 = Z23 .

Example 3.2. Determination of group-subgroupoids of group-groupoid Z12 (4). We apply the above method in the case of group-groupoid Z12 (4) of type (12; 3). Consider the group-groupoid Z12 (4) = (Z12 , α4 , β4 , 4 , ι4 , +, G0 ), where:

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Z12 = {c1 = 0, c2 = 4, c3 = 8, c4 = 1, c5 = 2, c6 = 3, c7 = 5, c8 = 6, c9 = 7, c10 = 9, c11 = 10, c12 = 11} and G0 = {c1 = 0, c2 = 4, c3 = 8}. The structure functions α4 , β4 : Z12 −→ G0 , ι4 : Z12 −→ Z12 and multiplication 4 : G(2) := {(x, y) ∈ Z12 × Z12 | 4x = 4y} −→ Z12 are given by (see, Example 2.2): α4 (x) = β4 (x) := 4x; ι4 (x) := 7x; x 4 y := 9x + y. Stage 1. We apply the program BGroidAP 2 for determining the set Hsgroid (Z12 (4)). The input data for the groupoid Z12 (4) of type (12; 3), are as follows:

12 3 1 1 1

2 2 2

3 3 3

2 2 9

3 3 5

1 1 10

3 1 2 3 1 2 12 8 4

1 1 6

2 2 11

3 3 7

1 0 0 2 0 0 0 4 0 0 6 0 0 0 8 0 0 9 10 0 0 11 0 0

0 0 3 0 5 0 7 0 0 0 0 12

0 4 0 11 0 0 0 0 2 0 9 0

0 0 5 0 3 0 12 0 0 0 0 7

6 0 0 0 0 8 0 10 0 1 0 0

0 0 7 0 12 0 5 0 0 0 0 3

8 0 0 0 0 10 0 1 0 6 0 0

0 9 0 2 0 0 0 0 11 0 4 0

10 0 0 0 0 1 0 6 0 8 0 0

The program BGroidAP 2 for these input data corresponding to groupoid Z12 (4), confirm the groupoid structure and displays the 63 subgroupoids. The subgroupoids of Z12 (4) containing {c1 } and whose order k is a divisor of n = 12, are as follows: 1 2 3 4 5 H(1;1) = {c1 }, H(2;1) = {c1 , c2 }, H(2;1) = {c1 , c3 }, H(2;1) = {c1 , c8 }, H(3;3) = {c1 , c2 , c3 }, 6 7 8 9 H(3;2) = {c1 , c2 , c8 }, H(3;2) = {c1 , c2 , c11 }, H(3;2) = {c1 , c3 , c5 }, H(3;2) = {c1 , c3 , c8 }, 10 11 12 13 H(4;3) = {c1 , c2 , c3 , c5 }, H(4;3) = {c1 , c2 , c3 , c8 }, H(4;3) = {c1 , c2 , c3 , c11 }, H(4;2) = {c1 , c2 , c8 , c11 }, 14 15 16 H(4;2) = {c1 , c3 , c5 , c8 }, H(4;2) = {c1 , c6 , c8 , c10 }, H(6;3) = {c1 , c2 , c3 , c4 , c9 , c11 }, 17 18 19 H(6;3) = {c1 , c2 , c3 , c5 , c7 , c12 }, H(6;3) = {c1 , c2 , c3 , c5 , c8 , c11 }, H(6;3) = {c1 , c2 , c3 , c6 , c8 , c10 }, 20 21 22 H(6;2) = {c1 , c2 , c4 , c8 , c9 , c11 }, H(6;2) = {c1 , c2 , c6 , c8 , c10 , c11 }, H(6;2) = {c1 , c3 , c5 , c6 , c8 , c10 }, 23 24 H(6;3) = {c1 , c3 , c5 , c7 , c8 , c12 } and H(12;3) = Z12 . Stage 2. We apply the program BGroidAP 2 to find the set Ksgr (Z12 ). The structure functions α0 , β0 : Z12 −→ {c1 } and ι0 : Z12 −→ Z12 , are given by α0 (x) = β0 (x) := c1 , ι0 (x) := −x, ∀ x ∈ Z12 . The input data for the group (Z12 , +) (groupoid of type (12; 1)) are as follows:

0 11 0 9 0 0 0 0 4 0 2 0

0 0 12 0 7 0 3 0 0 0 0 5


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1 2 3 4 5 6 7 8 9 10 11 12 2 3 1 7 8 9 10 11 12 4 5 6 3 1 2 10 11 12 4 5 6 7 8 9 4 7 10 5 6 2 8 9 3 11 12 1 12 5 8 11 6 2 7 9 3 10 12 1 4 1 6 9 12 2 7 8 3 10 11 1 4 5 1 1 1 1 1 1 1 1 1 1 1 1 7 10 4 8 9 3 11 12 1 5 6 2 1 1 1 1 1 1 1 1 1 1 1 1 8 11 5 9 3 10 12 1 4 6 2 7 1 3 2 12 11 10 9 8 7 6 5 4 9 12 6 3 10 11 1 4 5 2 7 8 10 4 7 11 12 1 5 6 2 8 9 3 11 5 8 12 1 4 6 2 7 9 3 10 12 6 9 1 4 5 2 7 8 3 10 11 The programm BGroidAP 2 for these input data corresponding to group (Z12 , +) confirm the group structure and displays the 6 subgroups, namely: 1 2 3 4 K(1;1) = {c1 }, K(2;1) = {c1 , c8 }, K(3;1) = {c1 , c2 , c3 }, K(4;1) = {c1 , c6 , c8 , c10 }, 5 6 K(6;1) = {c1 , c2 , c3 , c5 , c8 , c11 } and K(12;1) = Z12 .

Stage 3. Computing the intersection of the sets Hsgroid (Z12 (4)) and Ksgr (Z12 ) and one 1 4 5 15 16 24 obtains the set Sgrsgroid (Z12 (4)) = {H(1;1) , H(2;1) , H(3;3) = G0 , H(4;2) , H(6;3) , H(9;3) }. Hence, the group-groupoid Z12 (4) has exactly six group-subgroupoids, namely: K1 = {0}, K2 = {0, 6}, K3 = {0, 4, 8}, K4 = {0, 3, 6, 9}, K5 = {0, 2, 4, 6, 8, 10}, K6 = Z12 . Remark 3.1. Let G be a group-groupoid. The program BGroidAP 2 is used only to determine all subgroupoids of G. Effectiveness of the above algorithm consists in that it allows us to find all group-subgroupoids of G. Acknowledgments. The authors are very grateful to the reviewers for their comments and suggestions.

References [1] R. Brown, Topology and Groupoids, BookSurge LLC, U.K., 2006. [2] R. Brown and C.B. Spencer, G-groupoids, crossed modules and the fundamental groupoid of a topological group, Proc. Kon. Nederl. Akad. Wet., 79 (1976), 296-302. [3] M. Degeratu, Gh. Ivan and M. Ivan, On the cyclic subgroupoids of a Brandt groupoid, Proceed. Int. Conf. Comp., Commun. Control (ICCCC 2006), Baile Felix-Oradea. Vol. I, Suppl. Issue (2006), 181 -186. [4] Gh. Ivan, Algebraic constructions of Brandt groupoids, Proceedings of the Algebra Symposium, ”Babe¸sBolyai” University, Cluj-Napoca, (2002), 69-90. [5] Gh. Ivan and G. Stoianov, The program BGroidAP 2 for determination of all subgroupoids of a Brandt groupoid, Analele Univ. de Vest, Timi¸soara, Seria Mat.-Inf., 42, No 1 (2004), 93-118. [6] M. Ivan, Agebraic properties of G− groupoids, ArXiv: 1512.09012v1 [math.GR] 30 Dec 2015, p. 1-11. [7] K. Mackenzie, Lie Groupoids and Lie Algebroids in Differential Geometry, London Math. Soc., Lecture Notes Series, 124, Cambridge Univ.Press., 1987.

Received 25 April 2017

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1 West University of Timis ˆ rvan, ¸oara, Department of Teaching Staff’s Training, Bd. V. Pa no. 4, 300223, Timis¸oara, Romania 2 Department of Mathematics and Computer Science, University of Oradea, str. Universitatii nr. 1, 410087 Oradea, Romania E-mail address: 1 [email protected], 2 [email protected]