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 Chebyshevskii Sb., 2015, Volume 16, Issue 3, Pages 276–284 (Mi cheb418)

Free commutative $g$-dimonoids

A. V. Zhuchok, Yu. V. Zhuchok

Department of Algebra and System Analysis, Luhansk Taras Shevchenko National University, Gogol square, 1, Starobilsk, 92703, Ukraine

Abstract: A dialgebra is a vector space equipped with two binary operations $\dashv$ and $\vdash$ satisfying the following axioms:
\begin{gather*} (D1)\quad (x\dashv y)\dashv z=x\dashv (y\dashv z),
(D2)\quad (x\dashv y)\dashv z=x\dashv (y\vdash z),
(D3)\quad (x\vdash y)\dashv z=x\vdash (y\dashv z),
(D4)\quad (x\dashv y)\vdash z=x\vdash (y\vdash z),
(D5)\quad (x\vdash y)\vdash z=x\vdash (y\vdash z). \end{gather*}
This notion was introduced by Loday while studying periodicity phenomena in algebraic $K$-theory. Leibniz algebras are a non-commutative variation of Lie algebras and dialgebras are a variation of associative algebras. Recall that any associative algebra gives rise to a Lie algebra by $[x, y] =xy-yx$. Dialgebras are related to Leibniz algebras in a way similar to the relationship between associative algebras and Lie algebras. A dialgebra is just a linear analog of a dimonoid. If operations of a dimonoid coincide, the dimonoid becomes a semigroup. So, dimonoids are a generalization of semigroups.
Pozhidaev and Kolesnikov considered the notion of a $0$-dialgebra, that is, a vector space equipped with two binary operations $\dashv$ and $\vdash$ satisfying the axioms $(D2)$ and $(D4)$. This notion have relationships with Rota-Baxter algebras, namely, the structure of Rota-Baxter algebras appearing on $0$-dialgebras is known.
The notion of an associative $0$-dialgebra, that is, a $0$-dialgebra with two binary operations $\dashv$ and $\vdash$ satisfying the axioms $(D1)$ and $(D5)$, is a linear analog of the notion of a $g$-dimonoid. In order to obtain a $g$-dimonoid, we should omit the axiom $(D3)$ of inner associativity in the definition of a dimonoid. Axioms of a dimonoid and of a $g$-dimonoid appear in defining identities of trialgebras and of trioids introduced by Loday and Ronco.
The class of all $g$-dimonoids forms a variety. In the paper of the second author the structure of free $g$-dimonoids and free $n$-nilpotent $g$-dimonoids was given. The class of all commutative $g$-dimonoids, that is, $g$-dimonoids with commutative operations, forms a subvariety of the variety of $g$-dimonoids. The free dimonoid in the variety of commutative dimonoids was constructed in the paper of the first author.
In this paper we construct a free commutative $g$-dimonoid and describe the least commutative congruence on a free $g$-dimonoid.
Bibliography: 15 titles.

Keywords: dimonoid, $g$-dimonoid, commutative $g$-dimonoid, free commutative $g$-dimonoid, semigroup, congruence.

Full text: PDF file (252 kB)
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UDC: 512.57, 512.579
MSC: 08B20, 20M10, 20M50, 17A30, 17A32
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Citation: A. V. Zhuchok, Yu. V. Zhuchok, “Free commutative $g$-dimonoids”, Chebyshevskii Sb., 16:3 (2015), 276–284

Citation in format AMSBIB
\Bibitem{ZhuZhu15} \by A.~V.~Zhuchok, Yu.~V.~Zhuchok \paper Free commutative $g$-dimonoids \jour Chebyshevskii Sb. \yr 2015 \vol 16 \issue 3 \pages 276--284 \mathnet{http://mi.mathnet.ru/cheb418} \elib{http://elibrary.ru/item.asp?id=24398937} 

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This publication is cited in the following articles:
1. Yurii V. Zhuchok, “Automorphisms of the endomorphism semigroup of a free commutative $g$-dimonoid”, Algebra Discrete Math., 21:2 (2016), 309–324
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