The Cauchy-type product of two arithmetic functions f and g on nonnegative integers is defined by (f • g)(k) := Σ _m=0^k C(k, m) f(m)g(k-m). We explore some algebraic properties of the aforementioned convolution, which is a fundamental characteristic of the identities involving the Bernoulli numbers, the Bernoulli polynomials, the power sums, the sums of products, and so forth.