The variety of near-rings is generated by its finite members. (Q706201)

Let \(\mathcal V\) be a variety of algebras. A \(\mathcal V\)-composition algebra, \(\mathbf N\), is an algebra in \(\mathcal V\) with an additional associative binary operation, \(\circ\), such that each \(k\)-ary (\(k=1,2,\dots\)) operation \(\omega\) that is defined for \(\mathcal V\) satisfies \(\omega(x_1,\dots,x_k)\circ x_{k+1}=\omega(x_1\circ x_{k+1},\dots,x_k\circ x_{k+1})\) for all \(x_1,\dots,x_k\), \(x_{k+1}\in{\mathbf N}\). Let \(\mathbf A\) be an algebra in \(\mathcal V\), and let \({\mathbf M}({\mathbf A})\) be the set of all functions from \(\mathbf A\) to itself. Then \({\mathbf M}({\mathbf A})\) becomes an algebra in \(\mathcal V\) if one defines the operations \(\omega\) on \({\mathbf M}({\mathbf A})\) as the pointwise applications of the operations \(\omega\) as they are defined on \(\mathbf A\). In addition, if \(f,g\in{\mathbf M}({\mathbf A})\), and one defines \(f\circ g\) by \((f\circ g)(a)=f(g(a))\), for all \(a\in{\mathbf A}\), then \({\mathbf M}({\mathbf A})\) becomes a \(\mathcal V\)-composition algebra, called the full function algebra on \(\mathbf A\). It is shown that if \(\mathcal V\) is a variety that is generated by a class \(\mathcal F\) of algebras, then the variety of \(\mathcal V\)-composition algebras is generated by the class of all full function algebras on direct products of finitely many copies of algebras in \(\mathcal F\) (Theorem 3.1). Two corollaries follow. If \(p\) is a prime, then the variety of near-rings is generated by the near-rings \({\mathbf M}({\mathbf G})\), where \(\mathbf G\) ranges over all finite \(p\)-groups (Corollary 3.2). If \(\mathcal V\) is a variety of algebras that is generated by its finite members, so is the variety of \(\mathcal V\)-composition algebras (Corollary 3.3). For zero-symmetric near-rings, a result similar to Corollary 3.2 is obtained (Corollary 6.3).