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Fundam. Prikl. Mat., 2009, Volume 15, Issue 7, Pages 165–177 (Mi fpm1276)  

This article is cited in 3 scientific papers (total in 3 papers)

Abelian and Hamiltonian groupoids

A. A. Stepanova, N. V. Trikashnaya

Far Eastern National University

Abstract: In this work, we investigate some groupoids that are Abelian algebras and Hamiltonian algebras. An algebra is Abelian if for every polynomial operation and for all elements $a,b,\bar c,\bar d$ the implication $t(a,\bar c)=t(a,\bar d)\Longrightarrow t(b,\bar c)=t(b,\bar d)$ holds. An algebra is Hamiltonian if every subalgebra is a block of some congruence on the algebra. R. J. Warne in 1994 described the structure of the Abelian semigroups. In this work, we describe the Abelian groupoids with identity, the Abelian finite quasigroups, and the Abelian semigroups $S$ such that $abS=aS$ and $Sba=Sa$ for all $a,b\in S$. We prove that a finite Abelian quasigroup is a Hamiltonian algebra. We characterize the Hamiltonian groupoids with identity and semigroups under the condition of Abelianity of these algebras.

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English version:
Journal of Mathematical Sciences (New York), 2010, 169:5, 671–679

Bibliographic databases:

UDC: 510.8+512.57

Citation: A. A. Stepanova, N. V. Trikashnaya, “Abelian and Hamiltonian groupoids”, Fundam. Prikl. Mat., 15:7 (2009), 165–177; J. Math. Sci., 169:5 (2010), 671–679

Citation in format AMSBIB
\Bibitem{SteTri09}
\by A.~A.~Stepanova, N.~V.~Trikashnaya
\paper Abelian and Hamiltonian groupoids
\jour Fundam. Prikl. Mat.
\yr 2009
\vol 15
\issue 7
\pages 165--177
\mathnet{http://mi.mathnet.ru/fpm1276}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=2745008}
\transl
\jour J. Math. Sci.
\yr 2010
\vol 169
\issue 5
\pages 671--679
\crossref{https://doi.org/10.1007/s10958-010-0068-x}
\scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-77956056103}


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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. N. V. Trikashnaya, “Groupoids with Primitive Normal and Additive Theories”, J. Math. Sci., 195:6 (2013), 857–864  mathnet  crossref
    2. V. L. Usoltsev, “O gamiltonovykh ternarnykh algebrakh s operatorami”, Chebyshevskii sb., 15:3 (2014), 100–113  mathnet
    3. V. L. Usoltsev, “O gamiltonovom zamykanii na klasse algebr s odnim operatorom”, Chebyshevskii sb., 16:4 (2015), 284–302  mathnet  elib
  • Фундаментальная и прикладная математика
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