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108
The structure of conjugacy closed loops
 Trans. Amer. Math. Soc
, 2000
"... Abstract. We study structure theorems for the conjugacy closed (CC) loops, a specific variety of Gloops (loops isomorphic to all their loop isotopes). These theorems give a description all such loops of small order. For example, if p and q are primes, p
Abstract  Cited by 20 (5 self)  Add to MetaCartAbstract. We study structure theorems for the conjugacy closed (CC) loops, a specific variety of Gloops (loops isomorphic to all their loop isotopes). These theorems give a description all such loops of small order. For example, if p and q are primes, p<q,andq − 1 is not divisible by p, then the only CCloop of order pq is the cyclic group of order pq. Foranyprimeq>2, there is exactly one nongroup CCloop in order 2q, and there are exactly three in order q 2. We also derive a number of equations valid in all CCloops. By contrast, every equation valid in all Gloops is valid in all loops. 1.
The Complexity of Computing over Quasigroups
, 1994
"... In [7] the notions of recognition by semigroups and by programs over semigroups were extended to groupoids. This led to a new characterization of the contextfree languages and the class SAC¹. In this paper, we investigate the classes of languages obtained when the groupoids are restricted to be ..."
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Cited by 11 (6 self)
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In [7] the notions of recognition by semigroups and by programs over semigroups were extended to groupoids. This led to a new characterization of the contextfree languages and the class SAC¹. In this paper, we investigate the classes of languages obtained when the groupoids are restricted to be quasigroups (i.e. the multiplication table forms a latin square). We prove that languages recognized by quasigroups are regular and that programs over quasigroups characterize NC¹. We introduce the notions of linear recognition by groupoids and by programs over groupoids, and characterize the linear contextfree languages and NL. Here again, when quasigroups are used, only regular languages and languages in NC¹ can be obtained. We also consider the problem of evaluating a wellparenthesized expression over a finite loop (a quasigroup with an identity). This problem is in NC¹ for any finite loop, and we give algebraic conditions for its completeness. In particular, we prove that it is sufficient that the loop be nonsolvable, extending a wellknown theorem of Barrington ([3]).
Diassociativity in conjugacy closed loops
 Comm. Algebra
"... Let Q be a conjugacy closed loop, and N(Q) its nucleus. Then Z(N(Q)) contains all associators of elements of Q. If in addition Q is diassociative (i.e., an extra loop), then all these associators have order 2. If Q is powerassociative and Q  is finite and relatively prime to 6, then Q is a group. ..."
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Cited by 11 (6 self)
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Let Q be a conjugacy closed loop, and N(Q) its nucleus. Then Z(N(Q)) contains all associators of elements of Q. If in addition Q is diassociative (i.e., an extra loop), then all these associators have order 2. If Q is powerassociative and Q  is finite and relatively prime to 6, then Q is a group. If Q is a finite nonassociative extra loop, then 16  Q. 1
The structure of Fquasigroups
 in preparation. M. K. KINYON AND J. D. PHILLIPS
"... Abstract. We solve a problem of Belousov which has been open since 1967: to characterize the loop isotopes of Fquasigroups. We show that every Fquasigroup has a Moufang loop isotope which is a central product of its nucleus and Moufang center. We then use the loop to reveal the structure of the as ..."
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Cited by 9 (5 self)
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Abstract. We solve a problem of Belousov which has been open since 1967: to characterize the loop isotopes of Fquasigroups. We show that every Fquasigroup has a Moufang loop isotope which is a central product of its nucleus and Moufang center. We then use the loop to reveal the structure of the associated Fquasigroup. 1.
Moufang symmetry I. Generalized Lie and MaurerCartan equations
, 2008
"... The differential equations for a local analytic Moufang loop are established. The commutation relations for the infinitesimal translations of the analytic Moufang are found. These commutation relations can be seen as a (minimal) generalization of the MaurerCartan equations. ..."
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Cited by 8 (8 self)
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The differential equations for a local analytic Moufang loop are established. The commutation relations for the infinitesimal translations of the analytic Moufang are found. These commutation relations can be seen as a (minimal) generalization of the MaurerCartan equations.
Loops and semidirect products
, 2000
"... A left loop (B, ·) is a set B together with a binary operation · such that (i) for each a ∈ B, the left translation mapping La: B → B defined by La(x) = a · x is a bijection, and (ii) there exists a twosided identity 1 ∈ B satisfying 1 · x = x · 1 = x for every x ∈ B. A right loop is similarly def ..."
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Cited by 7 (4 self)
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A left loop (B, ·) is a set B together with a binary operation · such that (i) for each a ∈ B, the left translation mapping La: B → B defined by La(x) = a · x is a bijection, and (ii) there exists a twosided identity 1 ∈ B satisfying 1 · x = x · 1 = x for every x ∈ B. A right loop is similarly defined, and a loop is both a right loop
Finite Bruck loops
"... Let X be a magma; that is X is a set together with a binary operation ◦ on X. For each x ∈ X we obtain maps R(x) and L(x) on X defined by R(x) : y ↦ → y ◦ x and L(x) : y ↦ → x ◦ y called right and left translation by x, respectively. A loop is a magma X with an identity 1 such that R(x) and L(x) are ..."
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Cited by 7 (3 self)
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Let X be a magma; that is X is a set together with a binary operation ◦ on X. For each x ∈ X we obtain maps R(x) and L(x) on X defined by R(x) : y ↦ → y ◦ x and L(x) : y ↦ → x ◦ y called right and left translation by x, respectively. A loop is a magma X with an identity 1 such that R(x) and L(x) are permutations of X for all x ∈ X. In essence loops are groups without the associative axiom. See [Br, Pf] for further discussion of basic properties of loops. Certain classes of loops have received special attention: A loop X is a (right) Bol loop if it satisfies the (right) Bol identity (Bol): (Bol) or equivalently (Bol2) ((z ◦ x) ◦ y) ◦ x = z ◦ ((x ◦ y) ◦ x). R(x)R(y)R(x) = R((x ◦ y) ◦ x). for all x, y, z ∈ X. In a Bol loop, the subloop 〈x 〉 generated by x ∈ X is a group. Thus we can define x −1 and the order x  of x to be, respectively, the inverse of x and the
Every diassociative Aloop is Moufang
 Proc. Amer. Math. Soc
"... Abstract. An Aloop is a loop in which every inner mapping is an automorphism. We settle a problem which had been open since 1956 by showing that every diassociative Aloop is Moufang. 1. ..."
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Cited by 7 (5 self)
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Abstract. An Aloop is a loop in which every inner mapping is an automorphism. We settle a problem which had been open since 1956 by showing that every diassociative Aloop is Moufang. 1.
Involutory decomposition of groups into twisted subgroups and subgroups
 J. Group Theory
, 2000
"... Gyrogroups are generalized groups modelled on the Einstein groupoid of all relativistically admissible velocities with their Einstein’s velocity addition as a binary operation. Einstein’s gyrogroup fails to form a group since it is nonassociative. The breakdown of associativity in the Einstein addit ..."
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Cited by 7 (2 self)
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Gyrogroups are generalized groups modelled on the Einstein groupoid of all relativistically admissible velocities with their Einstein’s velocity addition as a binary operation. Einstein’s gyrogroup fails to form a group since it is nonassociative. The breakdown of associativity in the Einstein addition does not result in loss of mathematical regularity owing to the presence of the relativistic effect known as the Thomas precession which, by abstraction, becomes an automorphism called the Thomas gyration. The Thomas gyration turns out to be the missing link that gives rise to analogies shared by gyrogroups and groups. In particular, it gives rise to the gyroassociative and the gyrocommuttive laws that Einstein’s addition possesses, in full analogy with the associative and the commutative laws that vector addition possesses in a vector space. The existence of striking analogies shared by gyrogroups