On determinability of some classes of medial quasigroups by their endomorphisms

Leibak, Alar; Puusemp, Peeter

doi:10.1088/1742-6596/532/1/012014

Cited by 2 publications

(7 citation statements)

References 8 publications

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“…Therefore c 1 x 2 is a λ 2 -eigenvector of c 3 . This proves (38). Furthermore, since L c 1 is non-degenerated linear endomorphism, we also deduce that 0 (40) implies (39).…”

Section: 2supporting

confidence: 58%

“…Proposition 8.2 shows the existence of a 'hidden' abelian group structure on the set of idempotents of a medial algebra. This fact also follows from Proposition 4.1 in the paper of A. Leibak and P Puusemp [38].…”

Section: The Quasigroup Structure Of Isospectral Medial Alegrbassupporting

confidence: 57%

“…This implies that Idm(A) is a commutative idempotent medial (CIM, for short) quasigroup with respect to the algebra multiplication. Then the classical Bruck-Murdoch-Toyoda theorem characterizes any medial quasigroup as an isotope of a certain abelian group G. The structure of a CIM quasigroups and their endomorphisms has been recently studied by A. Leibak and P. Puusemp [38], [37]. In [38] a refinement of the Bruck-Murdoch-Toyoda theorem for CIM quasigroups has also been established.…”

Section: Resultsmentioning

confidence: 99%

“…Then the classical Bruck-Murdoch-Toyoda theorem characterizes any medial quasigroup as an isotope of a certain abelian group G. The structure of a CIM quasigroups and their endomorphisms has been recently studied by A. Leibak and P. Puusemp [38], [37]. In [38] a refinement of the Bruck-Murdoch-Toyoda theorem for CIM quasigroups has also been established. A natural question arise: Given a medial isospectral algebra A, how to characterize the underlying abelian group G?…”

Section: Resultsmentioning

confidence: 99%

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Medial and isospectral algebras

Krasnov¹,

Tkachev²

2022

Preprint

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The purpose of this paper is to give a systematic study of two new classes of commutative nonassociative algebras, the so-called isospectral and medial algebras. An isospectral algebra A is a generic commutative nonassociative algebra whose idempotents have the same Peirce spectrum. A medial algebra is algebra with identity (xy)(zw) = (xz)(yw). We show that these two classes are essentially coincide. We also prove that any medial spectral algebra is isomorphic to a certain isotopic deformation of the commutative associative quotient algebra K[z]/(z n − 1).Medial magma, Isospectral algebras, Quasigroups, Idempotents, Peirce decomposition

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“…Therefore c 1 x 2 is a λ 2 -eigenvector of c 3 . This proves (38). Furthermore, since L c 1 is non-degenerated linear endomorphism, we also deduce that 0 (40) implies (39).…”

Section: 2supporting

confidence: 58%

Section: The Quasigroup Structure Of Isospectral Medial Alegrbassupporting

confidence: 57%

Section: Resultsmentioning

confidence: 99%

Section: Resultsmentioning

confidence: 99%

See 2 more Smart Citations

Medial and isospectral algebras

Krasnov¹,

Tkachev²

2022

Preprint

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“…In fact, a homogeneous ccm-magma is the same as a commutative medial quasigroup (see for instance [16]). …”

Section: Internal Monoids and Internal Groupsmentioning

confidence: 99%

Internal monoids and groups in the category of commutative cancellative medial magmas

Fatelo¹,

Martins-Ferreira²

2016

Port. Math.

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Abstract. This article considers the category of commutative medial magmas with cancellation, a structure that generalizes midpoint algebras and commutative semigroups with cancellation. In this category each object admits at most one internal monoid structure for any given unit. Conditions for the existence of internal monoids and internal groups, as well as conditions under which an internal reflexive relation is a congruence, are studied.Mathematics Subject Classification (2010). Primary 08C15; Secondary 20N02.

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On determinability of some classes of medial quasigroups by their endomorphisms

Cited by 2 publications

References 8 publications

Medial and isospectral algebras

Medial and isospectral algebras

Internal monoids and groups in the category of commutative cancellative medial magmas

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