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On a certain algebra on a set of graphs. (English. Russian original) Zbl 0857.17002

Sib. Math. J. 35, No. 4, 706-712 (1994); translation from Sib. Mat. Zh. 35, No. 4, 793-800 (1994).
A bush over a set \(\Gamma\) is a pair \((A,\alpha)\) where \(A\) is a rooted tree and \(\alpha\) is a mapping from the set of vertices of \(A\) into \(\Gamma\). For two bushes \(A\), \(B\) over \(\Gamma\) and a vertex \(b\) of \(B\) denote by \(A(b)B\) the bush obtained by joining the root \(a_0\) of \(A\) to \(b\) by an edge.
Let \(k\) be a commutative associative ring with identity and without zero divisors. Let \(K\) be the set of elements of a free left \(k\)-module that is generated by all classes of isomorphic finite bushes over \(\Gamma\). Define a bilinear multiplication on \(K\) by putting \(AB=\sum_b A(b)B\) where the sum is over all vertices of \(B\).
It is proved that the algebra \(K\) satisfies the identity \(X(YZ)-(XY)Z- Y(XZ)+(YX)Z=0\); it is free in the variety of algebras satisfying the above identity, and has no zero divisors. A set of generators of \(K\) is indicated.
Reviewer: M.Demlová (Praha)

MSC:

17A30 Nonassociative algebras satisfying other identities
16D99 Modules, bimodules and ideals in associative algebras
17A60 Structure theory for nonassociative algebras
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[1] K. A. Zhevlakov, A. M. Slin’ko, I. P. Shestakov, and A. I. Shirshov, Rings That are Nearly Associative [in Russian], Nauka, Moscow (1978). · Zbl 0445.17001
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