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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
noncommutative variation of Lie algebras and dialgebras are a variation of associative
algebras. Recall that any associative algebra gives rise to a Lie algebra by
$[x, y] =xyyx$. 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 RotaBaxter algebras, namely, the structure of RotaBaxter 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.
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UDC:
512.57, 512.579
MSC: 08B20, 20M10, 20M50, 17A30, 17A32 Received: 01.07.2015
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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 276284
\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:

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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