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The Structure of Digroups AMERICAN JOURNAL OF UNDERGRADUATE RESEARCH VOL. 5, NO. 2 The Structure of Digroups Catherine Crompton Department of Mathematics Agnes Scott College 141 E. College Avenue Decatur, Georgia 30030 USA Linda Scalici Department of Mathematics Bucknell University Moore Avenue Lewisburg, Pennsylvania 17837 USA Received: November 7, 2005 Accepted: August 28, 2006 ABSTRACT A digroup is an algebra defined on a set having two associative binary operations, ⊢ and ⊣. Digroups play an important role in an open problem in the theory of Leibniz algebras. We present a brief overview of digroups and a set of more general axioms for a digroup than used previously. We then consider several properties of a digroup having distinct elements a and b such that a ⊢ b = b ⊢ a, but a ⊢ b ≠ a ⊣ b. I. INTRODUCTION Loday was the first to use the idea of a digroup [3]; since then others have A vexing problem in the theory of refined Loday’s work, e.g., [4] and [5]. Leibniz algebra is finding a generalization of Kinyon modified Loday’s definition of Lie’s third theorem. Lie’s third theorem digroup to give one that is much cleaner associates a (local) Lie group to any Lie (see below). As part of his partial solution to algebra, real or complex. Seeking the coquecigrue problem, he showed that appropriate analogues of Lie groups for every digroup is a product of a group and a Leibiniz algebras, Loday [1] became so trivial digroup. J. D. Phillips [6] gave an exasperated at one point that he dubbed even simpler basis of axioms for a digroup. these objects coquecigrues—absurdity In this paper, we generalize Kinyon’s first incarnate. Kinyon [2] found a partial solution axiom of associativity to give yet a more using digroups (defined below), as follows: if general axiom scheme for digroups. Then a Leibniz algebra splits over an ideal we look at some properties of pairs of containing the ideal generated by squares, elements that commute under one of the then it is isomorphic to the tangent algebra binary operations in a digroup. of a linear Lie digroup. In particular, if the Though we will not make use of this split Leibniz algebra is actually a Lie definition in this paper, a Lie digroup is a algebra, then the linear Lie digroup smooth manifold with a digroup structure coincides with the Lie group from Lie’s third such that the digroup operations are smooth theorem. For Leibniz algebras that do not mappings. A linear Lie digroup is a Lie split, the problem remains open. digroup recovered from the product of a Lie group and a module in a natural way [2]. 21 AMERICAN JOURNAL OF UNDERGRADUATE RESEARCH VOL. 5, NO. 2 II. AXIOMIZATIONS OF DIGROUPS Semi-Moufang identity. Kinyon’s definition of a digroup is as Theorem 2. A set, G, is a digroup if and follows. only if it is closed under two binary operations and , has a unary operation, Definition 1 (Kinyon). A digroup is a set, G, ⊢ ⊣ † and with nullary operation, 1, satisfying equipped with two binary operations, and , ⊢ the following axioms: ⊣, with a unary operation, †, and with nullary operation, 1, satisfying each of the following G1* ⊢ and ⊣ are semi-Moufang, i.e., six axioms: x ⊢ ((y ⊢ z) ⊢ x) = (x ⊢ y) ⊢ (z ⊢ x) and G1. (G, ⊢) and (G, ⊣) are both semigroups. x ⊣ ((y ⊣ z) ⊣ x) = (x ⊣ y) ⊣ (z ⊣ x). G2. (x ⊢ y) ⊣ z = x ⊢ (y ⊣ z). G2. (x ⊢ y) ⊣ z = x ⊢ (y ⊣ z). G3. x ⊣ (y ⊢ z) = x ⊣ y ⊣ z. G5. 1 ⊢ x = x = x ⊣ 1. † † G4. (x ⊣ y) ⊢ z = x ⊢ y ⊢ z. G6. x ⊢ x = 1 = x ⊣ x. G5. 1 ⊢ x = x = x ⊣ 1. Proof. All digroups satisfy these axioms † † because they are a weaker set of axioms. G6. x ⊢ x = 1 = x ⊣ x. Conversely, let G be an algebra satisfying Note that a digroup is essentially a left group these axioms. Using Phillips’ Axioms of a and a right group together with some digroup, we need only prove associativity of compatibility axioms. J. D. Phillips [6] the two binary operations, (since G2 implies showed that G2 through G4 can be G2* by setting y =x). We will show the proof simplified as follows: for the associativity of ⊢; the proof for ⊣ is similar. Theorem 1 (Phillips). A set, G, is a digroup, We introduce six auxiliary identities if and only if it is equipped with two binary to help in proving (x ⊢ y) ⊢ z = x ⊢ (y ⊢ z): operators, ⊢ and ⊣, with a unary operation, †, and with nullary operation, 1, satisfying A1. (x y) 1 = x (y 1). the following four axioms: ⊢ ⊢ ⊢ ⊢ A2. 1 ⊣ x = x ⊢ 1. † G1. (G, ⊢) and (G, ⊣) are both semigroups. A3. x ⊢ ((x ⊢ y) ⊢ x) = y ⊢ x † † G2* x ⊢ (x ⊣ z) = (x ⊢ x) ⊣ z. A4. x ⊢ 1 = x . † G5. 1 ⊢ x = x = x ⊣ 1. A5. x ⊢ (x ⊢ (y ⊢ 1)) = y ⊢ 1. † † G6. x ⊢ x = 1 = x ⊣ x. A6. (x ⊢ 1) ⊢ y = x ⊢ y. We first show that G must satisfy A1-A6 We will show that G1, the associativity of ⊢ then show that is associative. and ⊣, may be weakened to a form of the ⊢ G5 G *1 G5 A1: (x ⊢ y) ⊢ 1 = 1 ⊢ [(x ⊢ y) ⊢ 1] = (1 ⊢ x) ⊢ (y ⊢ 1) = x ⊢ (y ⊢ 1). G6 G2 G6 † † A2: 1 ⊣ x = (x ⊢⊢ x ) ⊣ x = = x ⊢ (x ⊣ x) = x ⊢ 1. G *1 G6 G5 † † A3: x ⊢ [(x ⊢ y) ⊢ x] = (x ⊢ x ) ⊢ (y ⊢ x) = 1 ⊢ (y ⊢ x) = y ⊢ x. G5 A1 A2 A2 † † † † † A4: x ⊢ 1 = x ⊢ (1 ⊢ 1) = (x ⊢ 1) ⊢ 1 = (1 ⊣ x ) ⊢ 1 = 1 ⊣ (1 ⊣ x ) G6 G *1 G5 G5 † † † † † † † † = (x ⊣ x) ⊣ (1 ⊣ x ) = x ⊣ [(x ⊣ 1) ⊣ x ] = x ⊣ (x ⊣ x ) = (x ⊣ 1) ⊣ (x ⊣ x ) G *1 A2 G2 A2 † † † † † † † † = x ⊣ [(1 ⊣ x) ⊣ x ] = x ⊣ [(x ⊢ 1) ⊣ x ] = x ⊣ [x ⊢ (1 ⊣ x )] = x ⊣ [x ⊢ (x ⊢ 1)] A1 G6 G5 † † † † = x ⊣ [(x ⊢ x ) ⊢ 1] = x ⊣ (1 ⊢ 1) = x . 22 AMERICAN JOURNAL OF UNDERGRADUATE RESEARCH VOL. 5, NO. 2 A2 G2 A2 † † † † A5: x ⊢ [x ⊢ (y ⊢ 1)] = x ⊢ [x ⊢ (1 ⊣ y)] = x ⊢ [(x ⊢ 1) ⊣ y] = x ⊢ [(1 ⊣ x ) ⊣ y] G5 G *1 G5 † † † = x ⊢ [(1 ⊣ x ) ⊣ (y ⊣ 1)] = x ⊢ [1 ⊣ (x ⊣ y)⊣ 1] = x ⊢ [1 ⊣ (x ⊣ y)] A2 A1 G2 G5 † † † = x ⊢ [(x ⊣ y) ⊢ 1] = [x ⊢ (x ⊣ y)] = [(x ⊢ x ) ⊣ y] ⊢ 1 = (1 ⊣ y) ⊢ 1 A2 A1 G5 = (y ⊢ 1) ⊢ 1 = y ⊢ (1 ⊢ 1) = y ⊢ 1. The following lengthy operations show that G satisfies A6. A3 A1 † † A6: (x ⊢ 1) ⊢ y = y ⊢ ((y ⊢ (x ⊢ 1)) ⊢ y) = y ⊢ (((y ⊢ x) ⊢ 1) ⊢ y) G *1 G5 A3 † † † † = (y ⊢ (y ⊢ x)) ⊢ (1 ⊢ y) = (y ⊢ (y ⊢ x)) ⊢ y = y ⊢ ((y ⊢ (y ⊢ (y ⊢ x))) ⊢ y) A4 † † = y ⊢ (((y ⊢ 1) ⊢ (y ⊢ (y ⊢ x))) ⊢ y) A5 † † † † † = y ⊢ ((((y ⊢ x) ⊢ ((y ⊢ x) ⊢ (y ⊢ 1))) ⊢ ((y ⊢ (y ⊢ 1))) ⊢ y) A4 † † † † † † = y ⊢ ((((y ⊢ x) ⊢ ((y ⊢ x) ⊢ y )) ⊢ (y y (y ⊢ x))) ⊢ y) G *1 † † † † † = y ⊢ (((y ⊢ x) ⊢ ((((y ⊢ x) ⊢ y ) ⊢ y) ⊢ (y ⊢ x))) ⊢ y) G5 † † † † † = y ⊢ (((y ⊢ x) ⊢ ((((y ⊢ x) ⊢ (1 ⊢ y )) ⊢ y) ⊢ (y ⊢ x))) ⊢ y) A4 † † † † † = y ⊢ (((y ⊢ x) ⊢ ((((y ⊢ x) ⊢ (1 ⊢ (y ⊢ 1))) ⊢ y) ⊢ (y ⊢ x))) ⊢ y) A2 † † † † † = y ⊢ (((y ⊢ x) ⊢ ((((y ⊢ x) ⊢ (1 ⊢ (1 ⊣ y ))) ⊢ y) ⊢ (y ⊢ x))) ⊢ y) G2 † † † † † = y ⊢ (((y ⊢ x) ⊢ ((((y ⊢ x) ⊢ ((1 ⊢ 1) ⊣ y )) ⊢ y) ⊢ (y ⊢ x))) ⊢ y) G2 † † † † † = y ⊢ (((y ⊢ x) ⊢ ((((y ⊢ x) ⊢ (1 ⊢ 1)) ⊣ y ) ⊢ y) ⊢ (y ⊢ x))) ⊢ y) A1 † † † † † = y ⊢ (((y ⊢ x) ⊢ ((((((y ⊢ x) ⊢ 1) ⊢ 1) ⊣ y ) ⊢ y) ⊢ (y ⊢ x))) ⊢ y) G2 † † † † † = y ⊢ (((y ⊢ x) ⊢ (((((y ⊢ x) ⊢ 1) ⊢ (1 ⊣ y )) ⊢ y) ⊢ (y ⊢ x))) ⊢ y) A2 † † † † † = y ⊢ (((y ⊢ x) ⊢ (((((y ⊢ x) ⊢ 1) ⊢ (y ⊢ 1)) ⊢ y) ⊢ (y ⊢ x))) ⊢ y) A4 † † † † † = y ⊢ (((y ⊢ x) ⊢ (((((y ⊢ x) ⊢ 1) ⊢ y ) ⊢ y) ⊢ (y ⊢ x))) ⊢ y) A3 † † † † † † = y ⊢ (((y ⊢ x) ⊢ ((y ⊢ ((y ⊢ (((y ⊢ x) ⊢ 1) ⊢ y )) ⊢ y)) ⊢ (y ⊢ x))) ⊢ y) A4 † † † † † † = y ⊢ (((y ⊢ x) ⊢ ((y ⊢ ((y ⊢ (((y ⊢ x) ⊢ 1) ⊢ (y ⊢ 1))) ⊢ y) ⊢ (y ⊢ x))) ⊢ y) A2 † † † † † † = y ⊢ (((y ⊢ x) ⊢ ((y ⊢ ((y ⊢ (((y ⊢ x) ⊢ 1) ⊢ (1 ⊣ y ))) ⊢ y) ⊢ (y ⊢ x))) ⊢ y) G2 † † † † † † = y ⊢ (((y ⊢ x) ⊢ ((y ⊢ ((y ⊢ ((((y ⊢ x) ⊢ 1) ⊢ 1) ⊣ y )) ⊢ y)) ⊢ (y ⊢ x))) ⊢ y) G2 † † † † † † = y ⊢ (((y ⊢ x) ⊢ ((y ⊢ (((y ⊢ (((y ⊢ x) ⊢ 1) ⊢ 1)) ⊣ y ) ⊢ y)) ⊢ (y ⊢ x))) ⊢ y) A1 † † † † † † = y ⊢ (((y ⊢ x) ⊢ ((y ⊢ ((((y ⊢ ((y ⊢ x) ⊢ 1)) ⊢ 1) ⊣ y ) ⊣ y)) ⊢ (y ⊢ x))) ⊢ y) 23 AMERICAN JOURNAL OF UNDERGRADUATE RESEARCH VOL. 5, NO. 2 G2 † † † † † † = y ⊢ (((y ⊢ x) ⊢ ((y ⊢ (((y ⊢ ((y ⊢ x) ⊢ 1)) ⊢ (1⊣ y )) ⊢ y)) ⊢ (y ⊢ x))) ⊢ y) A2 † † † † † † = y ⊢ (((y ⊢ x) ⊢ ((y ⊢ (((y ⊢ ((y ⊢ x) ⊢ 1)) ⊢ (y ⊢ 1)) ⊢ y)) ⊢ (y ⊢ x))) ⊢ y) A4 † † † † † † = y ⊢ (((y ⊢ x) ⊢ ((y ⊢ (((y ⊢ ((y ⊢ x) ⊢ 1)) ⊢ y ) ⊢ y)) ⊢ (y ⊢ x))) ⊢ y) G *1 † † † † † † = y ⊢ (((y ⊢ x) ⊢ (((y ⊢ (y ⊢ ((y ⊢ x) ⊢ 1))) ⊢ (y ⊢ y)) ⊢ (y ⊢ x))) ⊢ y) A5 † † † † † = y ⊢ (((y ⊢ x) ⊢ ((((y ⊢ x) ⊢ (y ⊢ y)) ⊢ (y ⊢ x))) ⊢ y) A4 † † † † † = y ⊢ (((y ⊢ x) ⊢ (((y ⊢ x) ⊢ (y ⊢ y)) ⊢ (y ⊢ x))) ⊢ y) A3 G *1 † † † † † = y ⊢ (((y ⊢ y) ⊢ (y ⊢ x)) ⊢ y) = (y ⊢ (y ⊢ y)) ⊢ ((y ⊢ y)) ⊢ ((y ⊢ x) ⊢ y) A4 A3 † † † = (y ⊢ ((y ⊢ 1) ⊢ y)) ⊢ ((y ⊢ x) ⊢ y) = (1 ⊢ y) ⊢ ((y ⊢ x) ⊢ y) G5 A3 † = y ⊢ ((y ⊢ x) ⊢ y) = x ⊢ y. We now show that ⊢ is associative. G2 = (x ⊢ ((1 ⊢ 1) ⊣ y)) ⊢ z x ⊢ (y ⊢ z) G2 A6 = (x ⊢ (1 ⊢ (1 ⊣ y))) ⊢ z = (x ⊢ 1) ⊢ (y ⊢ z) A2 A6 = (x ⊢ (1 ⊢ (y ⊢ 1))) ⊢ z = (x ⊢ 1) ⊢ ((y ⊢ 1) ⊢ z) G5 A5 = (x ⊢ (y ⊢ 1) ⊢ z † = (z ⊢ (z ⊢ (x ⊢ 1)) ⊢ ((y ⊢ 1) ⊢ z) A1 G *1 = ((x ⊢ y) ⊢⊢ 1) ⊢ z † = z ⊢ (((z ⊢ (x ⊢ 1)) ⊢ (y ⊢ 1)) ⊢ z) A6 A2 = (x ⊢ y) ⊢ z. † = z ⊢ (((z ⊢ (x ⊢ 1)) ⊢ (1 ⊣ y)) ⊢ z) □ G2 † = z ⊢ (((z ⊢ (x ⊢ 1)) ⊣ y) ⊢ z) III. COMMUTING PAIRS UNDER ⊢ A1 † = z ⊢ (((z ⊢ ((x ⊢ 1) ⊢ 1)) ⊣ y) ⊢ z) a. Example G2 = z ((z† (((x 1) 1) y)) z) ⊢ ⊢ ⊢ ⊣ ⊢ Suppose G is a digroup that is not a G2 † group, and further suppose that G has = z ⊢ ((z ⊢ ((x ⊢ 1) ⊢ (1 ⊣ y))) ⊢ z) A2 distinct elements a and b such that a ⊢ b = † = z ⊢ ((z ⊢ ((x ⊢ 1) ⊢ (y ⊢ 1))) ⊢ z) b ⊢ a and a ⊢ b ≠ a ⊣ b.
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