FREE COMMUTATIVE g-DIMONOIDS

Abstract

A dialgebra is a vector space equipped with two binary operations ⊣ and ⊢ satisfying the following axioms: (D1) (x ⊣ y) ⊣ z = x ⊣ (y ⊣ z), (D2) (x ⊣ y) ⊣ z = x ⊣ (y ⊢ z), (D3) (x ⊢ y) ⊣ z = x ⊢ (y ⊣ z), (D4) (x ⊣ y) ⊢ z = x ⊢ (y ⊢ z), (D5) (x ⊢ y) ⊢ z = x ⊢ (y ⊢ z). 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 anyassociativealgebragivesrisetoaLiealgebraby[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 ⊣ and ⊢ 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 ⊣ and ⊢ 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 shouldomittheaxiom (D3) ofinnerassociativityinthedefinitionofadimonoid. 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.