Permutation representations of left quasigroups

Smith, Jonathan D. H.

doi:10.1007/s00012-006-2005-x

Cited by 7 publications

(6 citation statements)

References 21 publications

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“…The category

{\underset{̲}{\underset{̲}{Q}}}_{fin}

of finite Q ‐ sets is the full subcategory of

{IFS}_{Q}

induced on the class of finite Q ‐sets. (Note that the alternative definitions here agree with the earlier definitions of [, ] — compare [, Th. 5.4].…”

Section: Iterated Function Systemssupporting

confidence: 80%

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Sylow Theory for Quasigroups

Smith

2014

J of Combinatorial Designs

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This paper is intended as a first step toward a general Sylow theory for quasigroups and Latin squares. A subset of a quasigroup lies in a nonoverlapping orbit if its respective translates under the elements of the left multiplication group remain disjoint. In the group case, each nonoverlapping orbit contains a subgroup, and Sylow's Theorem guarantees nonoverlapping orbits on subsets whose order is a prime‐power divisor of the group order. For the general quasigroup case, the paper investigates the relationship between non‐overlapping orbits and structural properties of a quasigroup. Divisors of the order of a finite quasigroup are classified by the behavior of nonoverlapping orbits. In a dual direction, Sylow properties of a subquasigroup P of a finite left quasigroup Q may be defined directly in terms of the homogeneous space P∖Q, and also in terms of the behavior of the isomorphism type [P∖Q] within the so‐called Burnside order, a labeled order structure on the full set of all isomorphism types of irreducible permutation representations.

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“…The category

{\underset{̲}{\underset{̲}{Q}}}_{fin}

of finite Q ‐ sets is the full subcategory of

{IFS}_{Q}

induced on the class of finite Q ‐sets. (Note that the alternative definitions here agree with the earlier definitions of [, ] — compare [, Th. 5.4].…”

Section: Iterated Function Systemssupporting

confidence: 80%

“…A Q ‐IFS is said to be a basic Q ‐ set if it is a homomorphic image of a homogeneous space

P ∖ Q

for a subquasigroup P of Q — compare Definition (a). Each basic Q ‐set is irreducible in the sense that it has no proper, nonempty subobjects , Cor. 8.2].…”

Section: Iterated Function Systemsmentioning

confidence: 99%

Sylow Theory for Quasigroups

Smith

2014

J of Combinatorial Designs

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“…ε-crisp coalgebras. In [10], Smith defines a Q-iterated function system (Q-IFS) as a Q-indexed family of stochastic linear maps on a vector space R(X). Since each linear map is determined by its restriction as Set-map α : X → R(X), a stochastic linear map is given by any mapping from X to the set of probability distributions on X, that is as a coalgebra of type D(X).…”

Section: Membership Through Constant Coalgebrasmentioning

confidence: 99%

“…Now the case of [10] is captured easily, as ε X : I Q → D Q where ε X (τ )(q) = τ (q), which inherits from x →x the property of being mono, natural, and sub-cartesian.…”

Section: Membership Through Constant Coalgebrasmentioning

confidence: 99%

On coalgebras and type transformations

Gumm

2007

Discuss. Math. - General Algebra and Applications

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Abstract. We show that for an arbitrary Set-endofunctor T the generalized membership function given by a sub-cartesian transformation µ from T to the filter functor F can be alternatively defined by the collection of subcoalgebras of constant T -coalgebras. Sub-natural transformations ε between any two functors S and T are shown to be sub-cartesian if and only if they respect µ. The class of T -coalgebras whose structure map factors through ε is shown to be a covariety if ε is a natural and sub-cartesian mono-transformation.

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“…There are some attempts to study permutation representations of quasigroups and loops-algebraic structures that are a generalization of groups. For instance, Jonathan Smith has intensively studied quasigroup and loop representations in a series of papers [9][10][11]. Also, the study of sharply transitive sets in quasigroup actions can be found in [7].…”

Section: Introductionmentioning

confidence: 99%

Gyrogroup actions: A generalization of group actions

Suksumran

2016

Journal of Algebra

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This article explores the novel notion of gyrogroup actions, which is a natural generalization of the usual notion of group actions. As a first step toward the study of gyrogroup actions from the algebraic viewpoint, we prove three well-known theorems in group theory for gyrogroups: the orbit-stabilizer theorem, the orbit decomposition theorem, and the Burnside lemma (or the Cauchy-Frobenius lemma). We then prove that under a certain condition, a gyrogroup G acts transitively on the set G/H of left cosets of a subgyrogroup H in G in a natural way. From this we prove the structure theorem that every transitive action of a gyrogroup can be realized as a gyrogroup action by left gyroaddition. We also exhibit concrete examples of gyrogroup actions from the Möbius and Einstein gyrogroups.

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Permutation representations of left quasigroups

Cited by 7 publications

References 21 publications

Sylow Theory for Quasigroups

Sylow Theory for Quasigroups

On coalgebras and type transformations

Gyrogroup actions: A generalization of group actions

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