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.
