On a certain algebra on a set of graphs (Q1920794)

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.