An external magma (S, \times) over R is a set S with an external binary operation.
An external semigroup (S, \times) over (R, \cdot) is an external magma that satisfies (r_1 \cdot r_2) \times s = r_1 \times (r_2 \times s) for all s \in S, r_1, r_2 \in R (externally associative).