Two theorems on identities in multioperator algebras

Two (unconnected) propositions on $\Omega$-algebras with identical relations are proved. The first of these (Theorem 1, in § 1) generalizes to $\Omega$-algebras a known fact from the theory of associative linear algebras, which asserts that every finite-dimensional algebra is an algebra with identical relations (more exactly, every algebra $A$ of dimension over a field $m$ satisfies a so-called standard identity of degree $m+1$). In § 2 we prove that every identical relation in an $\Omega$-algebra over a field of characteristic zero is equivalent to a system of polylinear identical relations (Theorem 2), from which it follows that the study of $\Omega$-algebras with arbitrary identical relations reduces to that of $\Omega$-algebras with polylinear identical relations. This theorem is proved in practically the same way as the corresponding proposition for ordinary algebras with identical relations, that is, algebras with a single binary multiplication (see for example, Mal'tsev [1]); it is clearly a generalization of it.