Free commutative $g$-dimonoids

A. V. Zhuchok ; Yu. V. Zhuchok

Čebyševskij sbornik, Tome 16 (2015) no. 3, pp. 276-284.

Résumé

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.

Détail

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

Bibliographie

[1] Pozhidaev A. P., “0-dialgebras with bar-unity and nonassociative Rota–Baxter algebras”, Sib. Math. J., 50:6 (2009), 1070–1080 | DOI | MR | Zbl

[2] Kolesnikov P. S., “Varieties of dialgebras and conformal algebras”, Sib. Math. J., 49:2 (2008), 257–272 | DOI | MR | Zbl

[3] Loday J.-L., “Dialgebras”, Dialgebras and related operads, Lecture Notes in Math., 1763, Springer-Verlag, Berlin, 2001, 7–66 | DOI | MR | Zbl

[4] Frabetti A., “Dialgebra (co)homology with coefficients”, Dialgebras and related operads, Lecture Notes in Math., 1763, Springer-Verlag, Berlin, 2001, 67–103 | DOI | MR | Zbl

[5] Bokut L. A., Chen Y., Liu C., “Gröbner–Shirshov bases for dialgebras”, Int. J. Algebra Comput., 20:3 (2010), 391–415 | DOI | MR | Zbl

[6] Kolesnikov P. S., Voronin V. Yu., “On the special identities for dialgebras”, Linear and Multilinear Algebra, 61:3 (2013), 377–391 | DOI | MR | Zbl

[7] Zhuchok A. V., “Dimonoids”, Algebra and Logic, 50:4 (2011), 323–340 | DOI | MR | Zbl

[8] Movsisyan Y., Davidov S., Safaryan Mh., “Construction of free $g$-dimonoids”, Algebra and Discrete Math., 18:1 (2014), 138–148 | MR | Zbl

[9] Zhuchok Yul. V., “On one class of algebras”, Algebra and Discrete Math., 18:2 (2014), 306–320 | MR | Zbl

[10] Loday J.-L., Ronco M. O., “Trialgebras and families of polytopes”, Contemp. Math., 346, 2004, 369–398 | DOI | MR | Zbl

[11] Casas J. M., “Trialgebras and Leibniz 3-algebras”, Boletín de la Sociedad Matemática Mexicana, 12:2 (2006), 165–178 | MR | Zbl

[12] Zhuchok A. V., “Semiretractions of trioids”, Ukr. Math. J., 66:2 (2014), 218–231 | DOI | MR | Zbl

[13] Zhuchok A. V., “Free commutative dimonoids”, Algebra and Discrete Math., 9:1 (2010), 109–119 | MR

[14] Zhuchok A. V., “Elements of dimonoid theory”, Mathematics and its Applications, Proceedings of Institute of Mathematics of NAS of Ukraine, 98, Kiev, 2014, 304 (in Ukrainian) | Zbl

[15] Zhuchok A. V., “Semilattices of subdimonoids”, Asian-Eur. J. Math., 4:2 (2011), 359–371 | DOI | MR | Zbl