On residual finiteness of direct products of algebraic systems (Q836913)

It is shown that the direct product \(S \times T\) of semigroups \(S\), \(T\) is residually finite iff both \(S\), \(T\) are residually finite. Simple counterexamples using algebras with idempotent unary operation show that in general residual finiteness of a direct product \(A \times B\) does not imply residual finiteness of universal algebras \(A\), \(B\).