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Free commutative \(g\)-dimonoids. (Russian. English summary) Zbl 1440.08001

Summary: 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 J.-L. Loday and M. Ronco [Contemp. Math. 346, 369–398 (2004; Zbl 1065.18007)].
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.

MSC:

08B20 Free algebras
08A30 Subalgebras, congruence relations
17A32 Leibniz algebras

Citations:

Zbl 1065.18007
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Full Text: MNR

References:

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