Left rays of groupoids

Asian-European Journal of Mathematics Vol. 16, No. 1 (2023) 2250233 (15 pages) c World Scientific Publishing Company  DOI: 10.1142/S1793557122502333 Left rays of groupoids A. Rezaei Department of Mathematics, Payame Noor University P. O. Box 19395-3697, Tehran, Iran rezaei@pnu.ac.ir H. S. Kim Department of Mathematics Research Institute of Natural Sciences Hanyang University, Seoul 04763, Korea heekim@hanyang.ac.kr J. Neggers Department of Mathematics, University of Alabama Tuscaloosa, AL 35487-0350, USA jneggers@ua.edu A. Borumand Saeid∗ Department of Pure Mathematics Faculty of Mathematics and Computer Shahid Bahonar University of Kerman, Kerman, Iran arsham@uk.ac.ir Communicated by P. Corsini Received December 5, 2021 Revised March 4, 2022 Accepted March 18, 2022 Published May 7, 2022 In this paper, we introduce and study the notion of a (positive implicative) left ray in groupoids, and we show that every normal subgroup of a group is a left ray of a group, and in every finite group, left rays are normal subgroups. Further, left absorptive subsets of groupoids are discussed and several examples for these definitions provided. The relationship between left rays and zero semigroups is discussed. Keywords: (Positive implicative) left ray; left absorptive; left ray; complete; left coset partition property; ray homomorphism; N -property. AMS Subject Classification: 20N02, 06F35 ∗ Corresponding author. 2250233-1 A. Rezaei et al. 1. Introduction and Preliminaries Given the category of groups and homomorphisms, normal subgroups are kernels of homomorphisms and the fundamental theorem of homomorphisms established an isomorphism between the image group and the quotient of the domain group by the kernel of the homomorphism (see [4, 6]). In establishing a theory of groupoids (binary systems) it is of interest to develop analogs of such a situation, which in the case of groups and homomorphisms reduces to the situation described above. In the situation described below that is the program we shall follow, i.e. we identify the kernel objects for given groupoids and from this identification we derive the proper class of associated mappings so that the image groupoid is “isomorphic” to a quotient groupoid involving the kernel object in a “natural” manner. Given that this “scheme” may potentially be developed in more than one way, we consider a theory of left rays of groupoids to be an example of such a theory, as is the theory of groups and normal subgroups as the kernel objects of group homomorphisms. Bruck [2] published a book, A Survey of Binary Systems discussed in the theory o of groupoids, loops and quasigroups, and several algebraic structures. Boruvka [1] stated the theory of decompositions of sets and its application to binary systems. Nebeský [12] introduced the notion of a travel groupoid by adding two axioms to a groupoid, and he described an algebraic interpretation of the graph theory. A groupoid (X, ∗) is said to be a right zero semigroup if x ∗ y = y for any x, y ∈ X, and a groupoid (X, ∗) is said to be a left zero semigroup if x ∗ y = x for any x, y ∈ X. A groupoid (X, ∗) is said to be a rightoid for ϕ : X → X if x ∗ y = ϕ(y) for any x, y ∈ X. Similarly, a groupoid (X, ∗) is said to be a leftoid for ϕ : X → X if x∗y = ϕ(x) for any x, y ∈ X. Note that a right (left, respectively) zero semigroup is a special case of a rightoid (leftoid, respectively). A groupoid (or semigroup) (X, ∗) is said to be right cancellative (or left cancellative, respectively) if y ∗ x = z ∗ x (x ∗ y = x ∗ z, respectively) implies y = z (see [3, 9]). Researchers proposed several kinds of algebraic structures related to some axioms in many-valued logic for investigation in many-valued logics. Y. Imai and K. Iséki introduced two classes of abstract algebras: BCK-algebras and BCI-algebras. The motivation of this notion is based on both set theory and propositional calculus (see [5, 7, 8, 10, 11]). As a generalization of these notions, the notion of d-algebras has been introduced by Neggers and Kim (see [13]). A d-algebra is a nonempty set X with a constant 0 and a binary operation “∗” satisfying the following axioms: (I) x ∗ x = 0, (II) 0 ∗ x = 0, (III) x ∗ y = 0 and y ∗ x = 0 imply x = y for all x, y ∈ X. In this paper, we introduce and study the notion of a left ray in groupoids. The relationship between left rays and normal subgroups is discussed, and we show that every normal subgroup of a group is a left ray of a group. Moreover, we show that if N is a left ray of X having the left coset partition property, then the mapping 2250233-2 Left rays of groupoids ϕ : (X, ∗) → (X/N, ⊛) is a homomorphism of groupoids. Further, the concept of left absorptive subsets of groupoids is defined and some new results are investigated. 2. Positive Implicative Left Rays In this section, we define positive implicative groupoids, and then consider positive implicative left rays of a groupoid. Also, left absorptive subsets of groupoids are discussed and several examples for these definitions provided. Definition 2.1. Let (X, ∗) be a groupoid. An element n ∈ X is said to be positive implicative if (x ∗ n) ∗ (y ∗ n) = (x ∗ y) ∗ n for all x, y ∈ X. Example 2.2. Let Z be the set of all integers. Define a binary operation “∗” on Z by x ∗ y := max{x, y − 1} for all x, y ∈ Z. Then every element of Z is a positive implicative element of Z. In fact, for any n ∈ Z, we have (x ∗ n) ∗ (y ∗ n) = max{x ∗ n, y ∗ n − 1} = max{max{x, n − 1}, max{y, n − 1} − 1} = max{max{x, n − 1}, max{y − 1, n − 2}} = max{x, y − 1, n − 1} = max{max{x, y − 1}, n − 1} = max{x ∗ y, n − 1} = (x ∗ y) ∗ n, which shows that n is positive implicative Example 2.3. Let K be a field with char(K) = 2. Define a binary operation “∗” on K by x ∗ y := 12 (x + y) for all x, y ∈ K. Then every element n ∈ K is positive y+n 1 implicative. In fact, for any x, y ∈ K, (x ∗ n) ∗ (y ∗ n) = 12 [ x+n 2 + 2 ] = 4 [x + y + 2n] and (x ∗ y) ∗ n = 12 [x + y + 2n]. Hence, n is positive implicative. Proposition 2.4. Let (X, ∗) be a right cancellative semigroup. If n is a positive implicative element of X, then n is a right identity of X. Proof. Since n is a positive implicative element of X, (x∗n)∗(y ∗n) = (x∗y)∗n for all x, y ∈ X. Since (X, ∗) is a semigroup, we have ((x ∗ n) ∗ y) ∗ n = (x ∗ n) ∗ (y ∗ n) = (x∗y)∗n. It follows from (X, ∗) is a right cancellative semigroup that (x∗n)∗y = x∗y, and hence x ∗ n = x for all x ∈ X, proving that n is a right identity of (X, ∗). Corollary 2.5. Let (X, ∗, e) be a group with identity e. If n is a positive implicative element of X, then n = e. 2250233-3 A. Rezaei et al. Proof. Assume (X, ∗, e) is a group and n is a positive implicative element of X. Since every group is a right cancellative semigroup, by Proposition 2.4, n is a right identity of the group (X, ∗, e), i.e. x ∗ n = x, for all x ∈ X. If put x := e, since e is an identity we get n = e ∗ n. On the other hand, since n is a right identity, we get e ∗ n = e, and so n = e. Definition 2.6. Let (X, ∗) be a groupoid. A nonempty subset N of X is said to be a positive implicative left ray of X if (LR1) (x ∗ N ) ∗ (y ∗ N ) ⊆ (x ∗ y) ∗ N for all x, y ∈ X, where x ∗ N := {x ∗ n | n ∈ N }. Note that if N = {n} is a positive implicative left ray of a groupoid, then n is a positive implicative element of the groupoid. Example 2.7. Let N := {1, 2, 3, . . .} be the set of all natural numbers and let “+” be the usual addition on N . If we define Aa := {a, a + 1, a + 2, . . .} where a ∈ N , then Aa is a positive implicative left ray of N . In fact, given x, y ∈ N, a ∈ N , we have (x + Aa ) + (y + Aa ) = {x + α|α ∈ Aa } + {y + β|β ∈ Aa } = {x + y + α + β|α, β ∈ Aa } ⊆ {x + y + γ|γ ∈ Aa } = (x + y) + Aa . Example 2.8. Let D4 := {ρ0 , ρ1 , ρ2 , ρ3 , µ1 , µ2 , δ1 , δ2 } be the fourth dihedral group [4]. If we take N := {ρ0 , ρ2 , δ1 , δ2 }, then it is easy to show that N is a positive implicative left ray of (D4 , ∗) and N C = {ρ1 , ρ3 , µ1 , µ2 }. We see that µ1 ∗ N C = {δ1 , δ2 , ρ0 , ρ2 } and δ1 ∗ N C = {µ1 , µ2 , ρ1 , ρ3 }. It follows that (µ1 ∗N C )∗(δ1 ∗N C ) = {ρ1 , ρ3 , µ1 , µ2 }. Since µ1 ∗δ1 = ρ3 , we obtain (µ1 ∗δ1 )∗N C = ρ3 ∗ N C = {ρ0 , ρ2 , δ1 , δ2 }. It follows that (µ1 ∗ N C ) ∗ (δ1 ∗ N C )  (µ1 ∗ δ1 ) ∗ N C . Hence, N C is not a positive implicative left ray of (D4 , ∗). Theorem 2.9. Let {Ni }ı∈I be a family of left rays of a groupoid (X, ∗). If N :=  i∈I Ni is nonempty, then it is a left ray of (X, ∗). The union of two positive implicative left rays of a groupoid need not be a positive implicative left ray of the groupoid. Example 2.10. In Example 2.8, if we take A := {ρ0 , µ1 } and B := {ρ0 , ρ2 }, then it is easy to see that they are positive implicative left rays of (D4 , ∗). Since A ∪ B = {ρ0 , ρ2 , µ1 } and µ1 ∗ δ2 = ρ1 , we have (µ1 ∗ δ2 ) ∗ (A ∪ B) = {ρ1 , ρ3 , δ1 }. 2250233-4 Left rays of groupoids It follows that µ1 ∗ (A ∪ B) = {µ1 , µ2 , ρ0 } and δ2 ∗ (A ∪ B) = {δ1 , δ2 , ρ3 }, and hence (µ1 ∗ (A ∪ B)) ∗ (δ2 ∗ (A ∪ B)) = {ρ1 , ρ3 , δ1 , δ2 }. This shows that (µ1 ∗ (A ∪ B)) ∗ (δ2 ∗ (A ∪ B)) = {ρ1 , ρ3 , δ1 , δ2 }  {ρ1 , ρ3 , δ1 } = (µ1 ∗ δ2 ) ∗ (A ∪ B), i.e. A ∪ B is not a positive implicative left ray of (D4 , ∗). The following example shows that every positive implicative left ray may not be a sub-algebra in general. Example 2.11 ([5]). Let X = {0, 1, 2, 3, 4}. We define the binary operation ∗ on X by Table 1. Table 1. Groupoid (X, ∗, 0). ∗ 0 1 2 3 4 0 1 2 3 4 0 1 2 3 4 0 0 2 3 4 0 1 0 3 4 0 0 2 0 2 0 0 0 0 0 Then (X, ∗, 0) is a positive implicative BCK-algebra and so a groupoid. If we take N := {3, 4}, then it is easy to check that N is a positive implicative left ray of X, but not a sub-algebra of X, since 4 ∗ 3 = 2 ∈ N. Definition 2.12. Let (X, ∗) be a groupoid. A nonempty subset N of X is said to be left absorptive if (LR2) x ∗ n = x ∗ m and n ∈ N imply m ∈ N for all x ∈ X. Example 2.13. In Example 2.8, it was proved that N C = {ρ1 , ρ3 , µ1 , µ2 } is not a positive implicative left ray of (D4 , ∗). Since the Dihedral 4-group (D4 , ∗) satisfies the cancellation laws, we see that N C is left absorptive. Proposition 2.14. Let (X, ∗) be a groupoid and let N be a left absorptive subset of (X, ∗). If X\N = ∅, then X\N is also a left absorptive subset of (X, ∗). Proof. Assume X\N is not a left absorptive subset of (X, ∗). Then there exist n ∈ X\N and m ∈ X\N such that x ∗ n = x ∗ m. It follows that m ∈ N such that x ∗ n = x ∗ m. Since N is left absorptive, we obtain n ∈ N , a contradiction. 2250233-5 A. Rezaei et al. Proposition 2.15. Let (X, ∗) be a groupoid and let N, M be left absorptive subsets of (X, ∗). Then N ∪ M and N ∩ M are left absorptive subsets of (X, ∗). Corollary 2.16. Let (X, ∗) be a groupoid and let N, M be left absorptive subsets of (X, ∗). Then N \M is a left absorptive subset of (X, ∗). Proof. Since N \M = N ∩ M C , by applying Proposition 2.15, we obtain N \M is a left absorptive subset of (X, ∗). 3. Left Rays of Groupoids In this section, we consider left rays (respectively, right rays) of groupoids, and new results and examples on these new concepts have been investigated. The relationship between left rays and normal subgroups are discussed. Definition 3.1. Let (X, ∗) be a groupoid. A nonempty subset N of X is said to be a left ray of X if it is both a left absorptive subset of X and a positive implicative left ray of X, i.e. (LR1) (x ∗ N ) ∗ (y ∗ N ) ⊆ (x ∗ y) ∗ N , for all x, y ∈ X, (LR2) x ∗ n = x ∗ m and n ∈ N imply m ∈ N , for all x ∈ X, where x ∗ N := {x ∗ n | n ∈ N }. The concept of right rays is a natural dual. Thus, a nonempty subset N of a groupoid (X, ∗) is said to be a right ray of a groupoid (X, ∗) if it satisfies the following conditions: (RR1) (N ∗ x) ∗ (N ∗ y) ⊆ N ∗ (x ∗ y), for all x, y ∈ X, (RR2) n ∗ x = m ∗ x and n ∈ N imply m ∈ N , for all x ∈ X, where N ∗ x := {n ∗ x | n ∈ N }. If (X, •) = (X, ∗)op where y • x = x ∗ y, then a left ray of (X, ∗) is a right ray of (X, •) = (X, ∗)op , while a right ray N of (X, ∗) is a left ray of (X, •) = (X, ∗)op . Note that if (X, ∗) is a commutative groupoid, i.e. x ∗ y = y ∗ x for all x, y ∈ X, then left rays coincide with right rays. Example 3.2. Let R be the set of all real numbers and let N := [α, ∞) where α ≥ 0. Then N is a left ray of (R, +). In fact, given x ∈ R, we have x + N = x + [α, ∞) = [x + α, ∞). For any x, y ∈ R, we have (x + N ) + (y + N ) = [x + α, ∞) + [y + α, ∞) = [x + y + 2α, ∞) 2250233-6 Left rays of groupoids = (x + y) + [α, 2∞) ⊆ (x + y) + [α, ∞) = (x + y) + N. Given x, m ∈ R, if n ∈ N with x + n = x + m, then m = n ∈ N . Hence, N is a left ray of (R, +). Example 3.3. Let (X, ∗, 0) be a d-algebra and let N be a left ray of X. Let n0 ∈ N . Given m ∈ X, since (X, ∗, 0) is a d-algebra, we have 0 ∗ m = 0 = 0 ∗ n0 . By (LR2), we obtain m ∈ N . This shows that N = X. Proposition 3.4. (i) Let (X, ∗) be a right zero semigroup. Then every nonempty subset N of X is a left ray of X. (ii) Let (X, ∗) be a left zero semigroup. If N is a left ray of X, then N = X. Proof. (i) Let N be a nonempty subset of X. Given x, y ∈ X and a, b ∈ N , we have (x∗a)∗(y ∗b) = x∗y = x = (x∗y)∗a ∈ (x∗y)∗N . Hence (x∗N )∗(y ∗N ) ⊆ (x∗y)∗N , i.e. N is a left ray of (X, ∗). (ii) Assume N is a left ray of (X, ∗) such that N = X. Let m ∈ X\N and n ∈ N . Since (X, ∗) is a left zero semigroup, we have x ∗ m = x = x ∗ n for all x ∈ X. Since n ∈ N and N is a left zero ray of X, we obtain m ∈ N , a contradiction. Proposition 3.5. Let (X, ∗, e) be a group and let g ∈ X. If N := {g} is a left ray of X, then g = e. Proof. Let N = {g} be a left ray of (X, ∗, e). Then (x ∗ N ) ∗ (y ∗ N ) ⊆ (x ∗ y) ∗ N for all x, y ∈ X, i.e. (x ∗ g) ∗ (y ∗ g) ∈ {(x ∗ y) ∗ g}. If we take x := e, y := e, then (e ∗ g) ∗ (e ∗ g) = (e ∗ e) ∗ g. Hence g 2 = g, proving that g = e. Theorem 3.6. Every normal subgroup of a group is a left ray of the group. Proof. If N is a normal subgroup of a group (X, ∗, e), then x ∗ N = N ∗ x for all x ∈ X. It follows that, for any x, y ∈ X, we have (x ∗ N ) ∗ (y ∗ N ) = x ∗ [N ∗ (y ∗ N )] = x ∗ [(N ∗ y) ∗ N ] = x ∗ [(y ∗ N ) ∗ N ] = x ∗ [y ∗ (N ∗ N )] ⊆ x ∗ (y ∗ N ) = (x ∗ y) ∗ N. Since every group has the cancellation laws, N is a left ray of a group (X, ∗, e). 2250233-7 A. Rezaei et al. Proposition 3.7. Let (X, ∗) be a leftoid for ϕ, i.e. x ∗ y := ϕ(x) for all x, y ∈ X. If N is a left ray of (X, ∗), then N = X. Proof. Given x, m ∈ X and n ∈ N , since N is a leftoid of (X, ∗), we have x ∗ n = ϕ(x) = x ∗ m. Since N is a left ray of X, we obtain m ∈ N , proving that N = X. Proposition 3.8. Let (X, ∗) be a rightoid for ϕ, i.e. x ∗ y = ϕ(y) for all x, y ∈ X. If N is a left ray of X, then ϕ(ϕ(N )) ⊆ ϕ(N ) and N = ϕ−1 (ϕ(N )). Proof. Let N be a left ray of (X, ∗). Then (x ∗ N ) ∗ (y ∗ N ) ⊆ (x ∗ y) ∗ N for all x, y ∈ X. It follows that ϕ(ϕ(N )) = ϕ(N ) ∗ ϕ(N ) = (x ∗ N ) ∗ (y ∗ N ) ⊆ (x ∗ y) ∗ N = ϕ(N ). If m ∈ ϕ−1 (ϕ(N )), then ϕ(m) ∈ ϕ(N ) and hence ϕ(m) = ϕ(n) for some n ∈ N . It follows that x ∗ m = ϕ(m) = ϕ(n) = x ∗ n for all x ∈ X. Since N is a left ray of (X, ∗) and n ∈ N , we obtain m ∈ N . Hence ϕ−1 (ϕ(N )) ⊆ N , proving that ϕ−1 (ϕ(N )) = N . Theorem 3.9. Let {Ni }ı∈I be a family of left rays of a groupoid (X, ∗). If N :=  i∈I Ni is nonempty, then it is a left ray of (X, ∗). Proof. (LR2) Given x ∈ X, if we let x ∗ n = x ∗ m and n ∈ N , then n ∈ Ni for all i ∈ I. Since Ni is a left ray of X, we obtain m ∈ Ni for all i ∈ I, i.e. m ∈ N . (LR1) Given x, y ∈ X, if α ∈ (x ∗ N ) ∗ (y ∗ N ), then there exist p, q ∈ N such that  α = (x ∗ p) ∗ (y ∗ q). Since N = i∈I Ni , we obtain α = (x ∗ p) ∗ (y ∗ q) ∈ (x ∗ Ni ) ∗ (y ∗ Ni ) ⊆ (x ∗ y) ∗ Ni , for all i ∈ I. It follows that there exists u ∈ Ni such that α = (x ∗ y) ∗ u. We claim  that u ∈ N = i∈I Ni . If u ∈ N , then there exists j ∈ I such that u ∈ Nj . Since α ∈ (x ∗ Nj ) ∗ (y ∗ Nj ) ⊆ (x ∗ y) ∗ Nj , there exists v = u such that α = (x ∗ y) ∗ v. It follows that (x ∗ y) ∗ u = α = (x ∗ y) ∗ v. Since v ∈ Nj , by applying (LR2), we obtain u ∈ Nj , a contradiction. Thus, α = (x ∗ y) ∗ u ∈ (x ∗ y) ∗ N . Hence (x ∗ N ) ∗ (y ∗ N ) ⊆ (x ∗ y) ∗ N , proving (LR1). Thus, N is a left ray of (X, ∗). Theorem 3.10. Let ϕ : (X, ∗) → (Y, ·) be a homomorphism of groupoids and N be a left ray of (X, ∗), and let M be a left ray of (Y, ·). Then (i) ϕ−1 (M ) is a left ray of (X, ∗), (ii) if ϕ is an epimorphism, then ϕ(N ) is a left ray of (Y, ·). 2250233-8 Left rays of groupoids Proposition 3.11. Let (X, ∗, e) be a finite group. If N is a left ray of (X, ∗), then it is a normal subgroup of (X, ∗, e). Proof. If N is a left ray of (X, ∗), then (e ∗ N )∗ (e ∗ N ) ⊆ (e ∗ e)∗ N , i.e. N ∗ N ⊆ N . It shows that N is a subsemigroup of X. Since X is finite, N is a subgroup of X. We claim that (N, ∗) is a normal subgroup of (X, ∗, e). Given x, y ∈ X, n1 , n2 ∈ N , by (LR1), there exists n ∈ N such that (x ∗ n1 ) ∗ (y ∗ n2 ) = (x ∗ y) ∗ n. (1) If we take x := e in (1), then there exists n3 ∈ N such that (e ∗ n1 ) ∗ (y ∗ n2 ) = (e ∗ y) ∗ n3 . It follows that (n1 ∗ y) ∗ n2 = n1 ∗ (y ∗ n2 ) = y ∗ n3 . −1 ′ ′ Hence, n1 ∗ y = (y ∗ n3 ) ∗ n−1 2 = y ∗ (n3 ∗ n2 ) = y ∗ n for some n ∈ N . Thus, we have N ∗ y ⊆ y ∗ N . By symmetry, we obtain y ∗ N ⊆ N ∗ y. Hence, y ∗ N = N ∗ y for all y ∈ X. This proves that N is a normal subgroup of X. A left ray N of a groupoid (X, ∗) is said to be complete if  (LR3) i∈I ai ∗ N = a ∗ N for some a ∈ X. We provide an example of a left ray which is not a subgroup as follows. Example 3.12. Let Q be the set of all rational numbers and let “ + ” be the usual addition on Q. Let P be the set of all natural numbers. Then, for all x, y ∈ Q, p1 , p2 ∈ P , we have (x + p1 ) + (y + p2 ) = (x + y) + (p1 + p2 ). It shows that (x + P ) + (y + P ) ⊆ (x + y) + P . Assume x + p1 = x + p2 and p2 ∈ P where x ∈ Q. Then, by cancellation law in (Q, +), we obtain p1 = p2 ∈ P . Hence, P is a left ray of (Q, +). Clearly, P is not a subgroup of (Q, +). We claim that P  is complete. If u ∈ i∈I (ai + P ), then u = ai + pi for all i ∈ I where pi ∈ P . It follows that u + P = (ai + pi ) + P = ai + (pi + P ) ⊆ ai + P,  for all i ∈ I. It shows that u + P ⊆ i∈I (ai + P ) ⊆ u + P , proving the claim. Remark 3.13. Note that in Proposition 3.11, the condition finite is necessary and so we cannot remove it. For this, see Examples 3.2 and 3.12. Remark 3.14. We give another proof of the completeness of P in Example 3.12 as  follows. First, we claim that if ai > aj , then ai + P ⊆ aj + P . Let u ∈ i∈I (ai + P ). Then there exist i, j ∈ I such that ai + pi = u = aj + pj where ai , aj ∈ Q and pi , pj ∈ P . It follows that ai = u − pi and aj = u − pj . Since ai > aj , we have 2250233-9 A. Rezaei et al. ai − aj = (u − pi ) − (u − pj ) = pj − pi ∈ P . Hence ai = aj + (pj − pi ) ∈ aj + P , and hence ai + P ⊆ aj + P , proving the claim. Since u = ai + pi for all i ∈ I, we obtain u > ai . By applying the claim, we have  u + P ⊆ ai + P for all i ∈ I. Hence, u + P ⊆ i∈I (ai + P ) ⊆ u + P . Theorem 3.15. Let N be a left ray of a groupoid (X, ∗) and let M be a left ray of a groupoid (Y, •). Define Z := X × Y and define (x, y)▽(u, v) := (x ∗ u, y • v) on Z. Then (N × M, ▽) is a left ray of X × Y . Proof. (LR1) Given (x, y), (u, v) ∈ Z, we have [(x, y) ▽ (N × M )]▽[(u, v) ▽ (N × M )] = [(x ∗ N ) × (y • M )] ▽ [(u ∗ N ) × (v • M )] = [(x ∗ N ) ∗ (u ∗ N )] × [(y • M ) • (v • M )] ⊆ [(x ∗ u) ∗ N ] × [(y • v) • M ] = (x ∗ u, y • v)▽(N × M ) = {(x, y)▽(u, v)}▽(N × M ). (LR2) Let (x, y)▽(n, m) = (x, y)▽(u, v) where (n, m) ∈ N × M and (u, v) ∈ Z. Then (x ∗ n, y • m) = (x ∗ u, y • v) and hence x ∗ n = y ∗ u and y • m = y • v. Since n ∈ N and N is a left ray of (X, ∗), and m ∈ M and M is a left ray of (Y, •), we obtain u ∈ N and v ∈ M . Hence, (u, v) ∈ N × M . 4. Quotient Structures Induced by Left Rays In this section, we will consider a left ray subset of a groupoid to construct related quotient structure. Let (X, ∗) be a groupoid. Let N be a left ray of (X, ∗) and let x, y ∈ X. Define a relation x ≡ y(mod N ) by x ∗ N = y ∗ N . Then ≡ is an equivalence relation on X. Let [x]N := {y ∈ X | x ∗ N = y ∗ N } and X/N := {[x]N | x ∈ X}. Define a binary relation “⊛” on X/N by where a(x,y) ∗ N :=  u∈[x]N v∈[y]N [x]N ⊛ [y]N := [a(x,y) ]N , (u ∗ v) ∗ N . Let x ≡ x′ (mod N ) and y ≡ y ′ (mod N ). Then [x]N = [x′ ]N and [y]N = [y ′ ]N . If we define [x]N ⊛ [y]N := [a(x,y) ]N and [x′ ]N ⊛ [y ′ ]N := [a′(x,y) ]N , then we have   a(x,y) ∗ N = (u ∗ v) ∗ N = (u ∗ v) ∗ N = a′(x,y) ∗ N. u∈[x]N v∈[y]N u∈[x′ ]N v∈[y ′ ]N It follows that [x]N ⊛ [y]N = [x′ ]N ⊛ [y ′ ]N . Hence ⊛ is well defined, i.e. (X/N, ⊛) is a groupoid. 2250233-10 Left rays of groupoids Example 4.1. Let X := {0, 1, 2, 3}. Define a binary operation ∗ on X by Table 2. Table 2. d-Algebra (X, ∗, 0). ∗ 0 1 2 3 0 1 2 3 0 1 2 2 0 0 2 2 0 1 0 1 0 0 1 0 Then (X, ∗, 0) is a d-algebra. By Example 3.3, we know that X is the only left ray of X. We denote it by N . Since 0∗X = {0}, 1∗X = {0, 1} and 2∗X = {0, 1, 2} = 3∗X, we obtain [0]N = {0}, [1]N = {1} and [2]N = [3]N = {2, 3}. Construct a quotient structure X/N = {[0]N , [1]N , [2]N } as in Table 3. Table 3. d-Algebra (X/N, ⊛, [0]N ). ⊛ [0]N [1]N [2]N [0]N [1]N [2]N [0]N [1]N [2]N [0]N [0]N [2]N [0]N [0]N [0]N This shows that the groupoid (X/N, ⊛, [0]N ) is also a d-algebra. Proposition 4.2. Let (X, ∗) be a left zero semigroup and let N be a left ray of X. Then (i) N = X, (ii) [x]N = {x} and X/N = {{x}|x ∈ X}, (iii) (X/N, ⊛) is a left zero semigroup. Proof. (i) Given m ∈ X, since X is a left zero semigroup, we have x∗n = x = x∗m for all x ∈ X and n ∈ N . By (LR2), we obtain m ∈ N , proving that N = X. (ii) If y ∈ [x]N , then x ∗ N = y ∗ N . Since X is a left zero semigroup, we obtain x = y. Hence [x]N = {x}, and so, we get X/N = {[x]N |x ∈ X} = {{x}|x ∈ X}. (iii) Given x, y ∈ X, we have [x]N ⊛ [y]N = [a(x,y) ]N = [x ∗ y]N = [x]N . Hence, (X/N, ⊛) is a left zero semigroup. Proposition 4.3. Let (X, ∗, e) be a finite group. If N is a left ray of X, then X/N = {x ∗ N | x ∈ X}. Proof. If N is a left ray of X, then N is a normal subgroup of X by Proposition 3.11. If y ∗ N = x ∗ N , then y = y ∗ e ∈ x ∗ N , and hence y = x ∗ n for some 2250233-11 A. Rezaei et al. n ∈ N . It shows that [x]N = {y ∈ X|x ∗ N = y ∗ N } = {y ∈ X|y = x ∗ n for some n} = x ∗ N. Hence, X/N = {[x]N | x ∈ X} = {x ∗ N | x ∈ X}. Definition 4.4. A left ray N of a groupoid (X, ∗) is said to have a left coset partition property if a ∗ N ⊆ b ∗ N , then a ∗ N = b ∗ N for all a, b ∈ X. Theorem 4.5. Let (X, ∗) be a groupoid and let N be a left ray of X having the left coset partition property. Then the mapping ϕ : (X, ∗) → (X/N, ⊛) defined by ϕ(x) := [x]N is a homomorphism of groupoids. Proof. Given x, y ∈ X, we claim that [x ∗ y]N = [a(x,y) ]N . If α ∈ [a(x,y) ]N , then  (u ∗ v) ∗ N. α ∗ N = a(x,y) ∗ N = u∈[x]N v∈[y]N It follows that α ∗ N ⊆ (u ∗ v)∗ N for all u ∈ [x]N , v ∈ [y]N . Since x ∈ [x]N , y ∈ [y]N , we have α ∗ N ⊆ (x ∗ y) ∗ N . By applying the left coset partition property, we obtain α ∗ N = (x ∗ y) ∗ N . Hence α ∈ [x ∗ y]N , i.e. [a(x,y) ]N ⊆ [x ∗ y]N . If β ∈ [x ∗ y]N , then β ≡ x ∗ y (mod N ), and hence β ∗ N = (x ∗ y) ∗ N . It follows that  (u ∗ v) ∗ N ⊆ (x ∗ y) ∗ N = β ∗ N. a(x,y) ∗ N = u∈[x]N v∈[y]N By the left coset partition property, we obtain a(x,y) ∗ N = β ∗ N , i.e. β ∈ [a(x,y) ]N . Hence, [x ∗ y]N ⊆ [a(x,y) ]N . Given x, y ∈ X, we obtain ϕ(x) ⊛ ϕ(y) = [x]N ⊛ [y]N = [a(x,y) ]N = [x ∗ y]N [∵ claim] = ϕ(x ∗ y) proving the theorem. Definition 4.6. A map ϕ : (X, ∗) → (Y, •) is said to have the N -property if there exists a nonempty subset N of X such that a ∗ N = b ∗ N if and only if ϕ(a) = ϕ(b) for all a, b ∈ X. Proposition 4.7. The mapping ϕ described in Theorem 4.5 has the N -property. 2250233-12 Left rays of groupoids Proof. Assume a ∗ N = b ∗ N where a, b ∈ X. Then [a]N = [b]N , and hence ϕ(a) = [a]N = [b]N = ϕ(b). Assume ϕ(a) = ϕ(b). Then [a]N = [b]N , and hence a ∗ N = b ∗ N . Hence, ϕ has the N -property. Definition 4.8. Let (X, ∗) and (Y, ·) be groupoids and let N be a nonempty subset of X. A map ϕ : (X, ∗) → (Y, ·) is said to have the subproduct property for N if (i) (Imϕ, ·) is a subgroupoid of (Y, ·), (ii) if ϕ(x) · ϕ(y) = ϕ(a(x,y) ), then a(x,y) ∗ N ⊆ (x ∗ y) ∗ N for all x, y ∈ X. By Theorem 4.5 and Proposition 4.7, the mapping ϕ in Theorem 4.5 has the N -property and the subproduct property for N . Definition 4.9. A mapping ϕ : (X, ∗) → (Y, ·) is said to be a ray homomorphism if there exists a nonempty subset N of X such that (i) ϕ has the N -property, (ii) ϕ has the subproduct property for N . Proposition 4.10. Let ϕ : (X, ∗) → (Y, ·) be a homomorphism of groupoids having the N -property. Then ϕ is a ray homomorphism. Proof. Let ϕ : (X, ∗) → (Y, ·) be a homomorphism of groupoids. Then ϕ(x ∗ y) = ϕ(x) · ϕ(y) for all x, y ∈ X. If we let a(x,y) := x ∗ y, then a(x,y) ∗ N = (x ∗ y) ∗ N , which proves (ii) of Definition 4.8. Clearly, (Imϕ, ·) is a subgroupoid of (Y, ·). Thus, ϕ has the subproduct property for N , and hence ϕ is a ray homomorphism. By Theorem 4.5 and Proposition 4.7, if N is a left ray having the left coset partition property, then ϕ : (X, ∗) → (X/N, ⊛) is a ray homomorphism. Theorem 4.11. Let ϕ : (X, ∗) → (Y, ·) be a ray epimorphism. If ϕ has the N property, then there exists a homomorphism of groupoids θ : (X/N, ⊛) → (Y, ·) such that the diagram in Fig. 1 commutes. ϕ // Y z== z zz ν zzθ z  z X/N X Fig. 1. Diagram of rays homomorphisms. Proof. Define θ : X/N → Y by θ([x]N ) := ϕ(x). Then it is well defined. In fact, if [x]N = [y]N , then x∗N = y ∗N . Since ϕ has the N -property, we obtain ϕ(x) = ϕ(y). We claim that θ ◦ ν = ϕ. For any x ∈ X, we have (θ ◦ ν)(x) = θ(ν(x)) = θ([x]N ) = ϕ(x). We claim that θ is a ray homomorphism. Given [x]N , [y]N ∈ X/N , we 2250233-13 A. Rezaei et al. obtain θ([x]N ⊛ [y]N ) = θ([a(x,y) ]N ) = ϕ(a(x,y) ) = ϕ(x) • ϕ(y) [because ϕ is a ray homomorphism] = θ(ν(x)) • θ(ν(y)) = θ([x]N ) • θ([y]N ) proving the theorem. Example 4.12. In Example 3.12, if we let N := P , then it is a left ray of (Q, +). We claim that a + N = b + N if and only if a = b where a, b, ∈ Q. In fact, assume a < b with a + N = b + N . Then a + 1 ∈ a + N = b + N , and hence there exists j ≥ 1 such that a + 1 = b + j. It follows that a = b + (j − 1) ≥ b, a contradiction. Using this claim, we see that N cannot have the left coset partition property. In fact, if we assume 7.5 + N = 5.5 + N , then we get 7.5 = 5.5, a contradiction. Moreover, using the claim, we obtain [x]N = {y | x + N = y + N } = {x} for all x ∈ Q. Given x, y ∈ Q, there exists a(x,y) ∈ Q such that [x]N ⊛ [y]N = [a(x,y) ]N . Since  [x]N = {x} for all x ∈ Q, we have a(x,y) + N = u∈[x]N (u + v) + N = (x + y) + N , v∈[y]N which implies [a(x,y) ]N = [x + y]N . Hence, [x]N ⊛ [y]N = [x + y]N . Define a map ϕ : (Q, +) → (Q/N, ⊛) by ϕ(x) := [x]N = {x}. Then ϕ(x) ⊛ ϕ(y) = [x]N ⊛ [y]N = [a(x,y) ]N = [x + y]N = ϕ(x + y) for all x, y ∈ Q. Hence, ϕ is a homomorphism of groupoids. We claim that ϕ has the N -property. In fact, if a + N = b + N , then we obtain a = b by the above claim. Hence ϕ(a) = ϕ(b). Suppose ϕ(a) = ϕ(b), then [a]N = [b]N and hence {a} = {b}. It follows that a = b and a + N = b + N . By Proposition 4.10, ϕ is a ray homomorphism. Remark 4.13. In Example 3.12, if we let N := {0}, then we have a + N = b + N if and only if a = b. Hence, [x]N = {y ∈ Q|x + N = y + N } = {x}. It follows that the map ϕ : (Q, +) → (Q/{0}, +) defined by ϕ(x) = {x} is a ray homomorphism and a homomorphism of groupoids. 5. Comment Although we have studied “kernels of morphisms” in a most general setting, involving the usual notions from universal algebras, as a particular kind of equivalence relation, the constructions given above correspond more closely to the notion of “normal” subobject (as in normal subgroup of a (finite) group identified as left ray, i.e. a subset N which satisfies both (LR1) and (LR2) conditions). This justifies consideration of left rays N of groupoids (X, ∗) as interesting subsets of (X, ∗) and of interest for further investigation. The special cases of left rays N = {n}, identity elements of (X, ∗) which behave not unlike left identity elements to some extent 2250233-14 Left rays of groupoids and which have an influence on the overall structure of the groupoids to which they belong. Acknowledgment We wish to thank the reviewers for excellent suggestions that have been incorporated into the paper. References o 1. O. Boruvka, Foundations of the Theory of Groupoids and Groups (John Wiley & Sons, New York, NY, USA, 1976). 2. R. H. Bruck, A Survey of Binary Systems (Springer, New York, 1971). 3. H. F. Fayoumi, Locally-zero groupoids and the center of Bin(X), Commun. Korean Math. Soc. 26(2) (2011) 163–168. 4. J. B. Fraleigh, A First Course in Abstract Algebra (Addison Wesley, 2003). 5. Y. Huang, BCI-Algebras (Science Press, Beijing, 2006). 6. T. W. 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